Few results in connection with sum and product theorems of relative $(p,q)$-$\varphi$ order, relative $(p,q)$-$\varphi$ type and relative $(p,q)$-$\varphi$ weak type of meromorphic functions with respect to entire functions

TL;DR

This paper explores the properties of sum and product operations on various relative growth measures of meromorphic functions compared to entire functions, focusing on their $(p,q)$-$\

Contribution

It introduces new results connecting sum and product theorems with relative $(p,q)$-$\varphi$ growth measures of meromorphic functions.

Findings

Established basic properties of relative $(p,q)$-$\varphi$ order.

Derived sum and product relations for relative $(p,q)$-$\varphi$ type.

Analyzed weak type behaviors in relation to entire functions.

Abstract

Orders and types of entire and meromorphic functions have been actively investigated by many authors. In the present paper, we aim at investigating some basic properties in connection with sum and product of relative $(p,q)$-$\varphi$ order, relative $(p,q)$-$\varphi$ type, and relative $(p,q)$-$\varphi$ weak type of meromorphic functions with respect to entire functions.

Equations

$$\begin{array}[]{c}\rho_{g}^{\left(p,q\right)}\left(f,\varphi\right)\\ \lambda_{g}^{\left(p,q\right)}\left(f,\varphi\right)\end{array}=\underset{r\rightarrow\infty}{\lim}\begin{array}[]{c}\sup\\ \inf\end{array}\frac{\log^{\left[p\right]}T_{g}^{-1}\left(T_{f}\left(r\right)\right)}{\log^{\left[q\right]}\varphi\left(r\right)}.$$

$$\begin{array}[]{c}\sigma_{g}^{\left(p,q\right)}\left(f,\varphi\right)\\ \overline{\sigma}_{g}^{\left(p,q\right)}\left(f,\varphi\right)\end{array}=\underset{r\rightarrow+\infty}{\lim}\begin{array}[]{c}\sup\\ \inf\end{array}\frac{\log^{\left[p-1\right]}T_{g}^{-1}\left(T_{f}\left(r\right)\right)}{\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\rho_{g}^{\left(p,q\right)}\left(f,\varphi\right)}}.$$

$$\begin{array}[]{c}\tau_{g}^{\left(p,q\right)}\left(f,\varphi\right)\\ \overline{\tau}_{g}^{\left(p,q\right)}\left(f,\varphi\right)\end{array}=\underset{r\rightarrow+\infty}{\lim}\begin{array}[]{c}\inf\\ \sup\end{array}\frac{\log^{\left[p-1\right]}T_{g}^{-1}\left(T_{f}\left(r\right)\right)}{\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\lambda_{g}^{\left(p,q\right)}\left(f,\varphi\right)}}.$$

$${\left[M_f\left(r\right)\right]}^n\le M_f\left(r^{\delta }\right)$$

$$T_f\left(r\right)\le \log M_f\left(r\right)\le 3T_f\left(2r\right).$$

$${\lambda }_{g_1}^{\left(p,q\right)}\left(f_1\pm f_2,\phi \right)\le \max \left\{{\lambda }_{g_1}^{\left(p,q\right)}\left(f_1,\phi \right),{\lambda }_{g_1}^{\left(p,q\right)}\left(f_2,\phi \right)\right\}.$$

$$T_{f_1}\left(r\right)\le T_{g_1}\left[{\exp }^{\left[p\right]}\left[\left({\lambda }_{g_1}^{\left(p,q\right)}\left(f_1,\phi \right)+\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]$$

$$i.e.,T_{f_1}\left(r\right)\le T_{g_1}\left[{\exp }^{\left[p\right]}\left[\left(\Delta +\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right].$$

$$T_{f_2}\left(r\right)\le T_{g_1}\left[{\exp }^{\left[p\right]}\left[\left({\lambda }_{g_1}^{\left(p,q\right)}\left(f_2,\phi \right)+\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]$$

$$i.e.,T_{f_2}\left(r\right)\le T_{g_1}\left[{\exp }^{\left[p\right]}\left[\left(\Delta +\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right].$$

$$T_{f_1\pm f_2}\left(r\right)\le 2\log M_{g_1}\left[{\exp }^{\left[p\right]}\left[\left(\Delta +\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]+O(1)$$

$$i.e.,T_{f_1\pm f_2}\left(r\right)\le 3\log M_{g_1}\left[{\exp }^{\left[p\right]}\left[\left(\Delta +\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right].$$

$$T_{f_1\pm f_2}\left(r\right)\le \frac{1}{3}\log {\left[M_{g_1}\left[{\exp }^{\left[p\right]}\left[\left(\Delta +\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]\right]}^9$$

$$i.e.,T_{f_1\pm f_2}\left(r\right)\le \frac{1}{3}\log M_{g_1}\left[{\left[{\exp }^{\left[p\right]}\left[\left(\Delta +\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]}^{\sigma }\right]$$

$$i.e.,T_{f_1\pm f_2}\left(r\right)\le T_{g_1}\left[2{\left[{\exp }^{\left[p\right]}\left[\left(\Delta +\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]}^{\sigma }\right].$$

$$i.e.,~{}\underset{r\rightarrow\infty}{\lim\inf}\frac{\log^{\left[p\right]}T_{g_{1}}^{-1}\left(T_{f_{1}\pm f_{2}}\left(r\right)\right)}{\log^{\left[q\right]}\varphi\left(r\right)}<\left(\Delta+\varepsilon\right)~{}.~{}\ \ \ \ \ \ \ \ \ \ \$$

$${\lambda }_{g_1}^{\left(p,q\right)}\left(f_1\pm f_2,\phi \right)\le \Delta =\max \left\{{\lambda }_{g_1}^{\left(p,q\right)}\left(f_1,\phi \right),{\lambda }_{g_1}^{\left(p,q\right)}\left(f_2,\phi \right)\right\}.$$

$${\rho }_{g_1}^{\left(p,q\right)}\left(f_1\pm f_2,\phi \right)\le \max \left\{{\rho }_{g_1}^{\left(p,q\right)}\left(f_1,\phi \right),{\rho }_{g_1}^{\left(p,q\right)}\left(f_2,\phi \right)\right\}.$$

$${\lambda }_{g_1\pm g_2}^{\left(p,q\right)}\left(f_1,\phi \right)\ge \min \left\{{\lambda }_{g_1}^{\left(p,q\right)}\left(f_1,\phi \right),{\lambda }_{g_2}^{\left(p,q\right)}\left(f_1,\phi \right)\right\}.$$

$$T_{g_k}\left[{\exp }^{\left[p\right]}\left[\left({\lambda }_{g_k}^{\left(p,q\right)}\left(f_1,\phi \right)-\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]\le T_{f_1}\left(r\right)\text{ where }k=1,2$$

$$i.e,T_{g_k}\left[{\exp }^{\left[p\right]}\left[\left(\Psi -\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]\le T_{f_1}\left(r\right)\text{ where }k=1,2$$

$$T_{g_1\pm g_2}\left[{\exp }^{\left[p\right]}\left[\left(\Psi -\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]\le 2T_{f_1}\left(r\right)+O(1)$$

$$i.e.,T_{g_1\pm g_2}\left[{\exp }^{\left[p\right]}\left[\left(\Psi -\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]\right]<3T_{f_1}\left(r\right).$$

$$\frac{1}{9}\log M_{g_1\pm g_2}\left[\frac{{\exp }^{\left[p\right]}\left[\left(\Psi -\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]}{2}\right]<T_{f_1}\left(r\right)$$

$$i.e.,\log M_{g_1\pm g_2}{\left[\frac{{\exp }^{\left[p\right]}\left[\left(\Psi -\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]}{2}\right]}^{\frac{1}{9}}<T_{f_1}\left(r\right)$$

$$i.e.,\log M_{g_1\pm g_2}\left[{\left(\frac{{\exp }^{\left[p\right]}\left[\left(\Psi -\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]}{2}\right)}^{\frac{1}{\sigma }}\right]<T_{f_1}\left(r\right)$$

$$i.e.,T_{g_1\pm g_2}\left[{\left(\frac{{\exp }^{\left[p\right]}\left[\left(\Psi -\epsilon \right){\log }^{\left[q\right]}\phi \left(r\right)\right]}{2}\right)}^{\frac{1}{\sigma }}\right]<T_{f_1}\left(r\right)$$

$${\lambda }_{g_1\pm g_2}^{\left(p,q\right)}\left(f_1,\phi \right)=\ge \Psi =\min \left\{{\lambda }_{g_1}^{\left(p,q\right)}\left(f_1,\phi \right),{\lambda }_{g_2}^{\left(p,q\right)}\left(f_1,\phi \right)\right\}.$$

$${\rho }_{g_1\pm g_2}^{\left(p,q\right)}\left(f_1,\phi \right)\ge \min \left\{{\rho }_{g_1}^{\left(p,q\right)}\left(f_1,\phi \right),{\rho }_{g_2}^{\left(p,q\right)}\left(f_1,\phi \right)\right\}.$$

Full text

AMS Subject Classification** $\left(2010\right)$ **: $30D20,30D30,30D35$

1. Introduction, Definitions and Notations

Let $f$ be an entire function defined in the open complex plane $\mathbb{C}.$ The maximum modulus function $M_{f}\left(r\right)$ corresponding to $f$ (see [12]) is defined on $\left|z\right|=r$ as $M_{f}\left(r\right)=\QTATOP{\max}{\left|z\right|=r}\left|f\left(z\right)\right|$. A non-constant entire function $f$ is said have the Property (A) if for any $\sigma>1$ and for all sufficiently large $r,$ $\left[M_{f}\left(r\right)\right]^{2}\leq M_{f}\left(r^{\sigma}\right)$ holds $\left(\text{see \cite[cite]{[\@@bibref{}{1}{}{}]}}\right)$. When $f$ is meromorphic, one may introduce another function $T_{f}\left(r\right)$ known as Nevanlinna’s characteristic function of $f$ (see [5, p.4]), playing the same role as $M_{f}\left(r\right).$ If $f$ is non-constant entire function, then its Nevanlinna’s characteristic function is strictly increasing and continuous and therefore there exists its inverse functions $T_{f}^{-1}(r)$ $:$ $\left(\left|f\left(0\right)\right|,\infty\right)$ $\rightarrow$ $\left(0,\infty\right)$ with $\underset{s\rightarrow\infty}{\lim}T_{f}^{-1}\left(s\right)=\infty.$

However, throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna theory of meromorphic functions which are available in [5, 9, 10, 11] and therefore we do not explain those in details. Now we define $\exp^{[k]}x=\exp\left(\exp^{[k-1]}x\right)$ and $\log^{[k]}x=\log\left(\log^{[k-1]}x\right)$ for $x\in[0,\infty)$ and $k\in\mathbb{N}$ where $\mathbb{N}$ be the set of all positive integers. We also denote $\log^{[0]}x=x,$ $\log^{[-1]}x=\exp x,$ $\exp^{[0]}x=x$ and $\exp^{[-1]}x=\log x.$ Further we assume that throughout the present paper $p$ and $q$ always denote positive integers.

Mainly the growth investigation of meromorphic functions has usually been done through its Nevanlinna’s characteristic function in comparison with those of exponential function. But if one is paying attention to evaluate the growth rates of any meromorphic function with respect to an entire function, the notions of relative growth indicators [8] will come. Extending this notion, Debnath et. al. [4] introduce the definition of relative $\left(p,q\right)$-th order and relative $\left(p,q\right)$-th lower order of a meromorphic function $f$ with respect to another entire function $g$ respectively in the light of index-pair ( detail about index-pair one may see [4, 6, 7] ). For details about it, one may see [4]. Extending this notion, recently Biswas [2] introduce the definitions of relative $\left(p,q\right)$-$\varphi$ order and the relative $\left(p,q\right)$-$\varphi$ lower order of a meromorphic function $f$ with respect to another entire function $g$ as follows:

Definition 1**.**

[2]** Let $\varphi$ $:$ $[0,+\infty)\rightarrow(0,+\infty)$ be a non-decreasing unbounded function. The relative $\left(p,q\right)$-$\varphi$ order and the relative $\left(p,q\right)$-$\varphi$ lower order of a meromorphic function $f$ with respect to an entire function $g$ are defined as

[TABLE]

If we consider $\varphi(r)=r$, then the above definition reduce to the definitions of relative $\left(p,q\right)$-th order and relative $\left(p,q\right)$-th lower order of a meromorphic $f$ with respect to an entire $g,$ introduced by Debnath et. al. [4].

If the relative $\left(p,q\right)$-$\varphi$ order and the relative $\left(p,q\right)$-$\varphi$ lower order of $f$ with respect to $g$ are the same, then $f$ is called a function of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g$. Otherwise, $f$ is said to be irregular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g$.

Now in order to refine the above growth scale, one may introduce the definitions of other growth indicators, such as relative $\left(p,q\right)$-$\varphi$ type and relative $\left(p,q\right)$-$\varphi$ lower type* *of entire or meromorphic functions with respect to another entire function which are as follows:

Definition 2**.**

[2]** Let $\varphi$ $:$ $[0,+\infty)\rightarrow(0,+\infty)$ be a non-decreasing unbounded function. The relative $\left(p,q\right)$-$\varphi$ type and the relative $\left(p,q\right)$-$\varphi$ lower type of a meromorphic function $f$ with respect to another entire function $g$ having non-zero finite relative $\left(p,q\right)$-$\varphi$ order $\rho_{g}^{\left(p,q\right)}\left(f,\varphi\right)$ are defined as :

[TABLE]

Analogously, to determine the relative growth of $f$ having same non zero finite relative $\left(p,q\right)$-$\varphi$ lower order with respect to $g$, one can introduce the definition of relative $\left(p,q\right)$-$\varphi$ weak type $\tau_{g}^{\left(p,q\right)}\left(f\right)$ and the growth indicator $\overline{\tau}_{g}^{\left(p,q\right)}\left(f\right)$ of $f$ with respect to $g$ of finite positive relative $\left(p,q\right)$-$\varphi$ lower order $\lambda_{g}^{\left(p,q\right)}\left(f\right)$ in the following way:

Definition 3**.**

[2]** Let $\varphi$ $:$ $[0,+\infty)\rightarrow(0,+\infty)$ be a non-decreasing unbounded function. The relative $\left(p,q\right)$-$\varphi$ weak type $\tau_{g}^{\left(p,q\right)}\left(f,\varphi\right)$ and the growth indicator $\overline{\tau}_{g}^{\left(p,q\right)}\left(f,\varphi\right)$ of a meromorphic function $f$ with respect to another entire function $g$ having non-zero finite relative $\left(p,q\right)$-$\varphi$ lower order $\lambda_{g}^{\left(p,q\right)}\left(f,\varphi\right)$ are defined as :

[TABLE]

If we consider $\varphi(r)=r$, then $\sigma_{g}^{\left(p,q\right)}\left(f,r\right)$ and $\tau_{g}^{\left(p,q\right)}\left(f,r\right)$ are respectively known as relative $\left(p,q\right)$-th type and relative $\left(p,q\right)$-th weak type of $f$ with respect to $g$. For details about relative $\left(p,q\right)$-th type, relative $\left(p,q\right)$-th weak type etc., one may see [3].

Here, in this paper, we aim at investigating some basic properties of relative $\left(p,q\right)$-$\varphi$ order, relative $\left(p,q\right)$-$\varphi$ type and relative $\left(p,q\right)$-$\varphi$ weak type of a meromorphic function with respect to an entire function under somewhat different conditions. Throughout this paper, we assume that all the growth indicators are all nonzero finite.

2. Lemmas

In this section we present some lemmas which will be needed in the sequel.

Lemma 1**.**

[1]** Let $f$ be an entire function which satisfies the Property (A) then for any positive integer $n$ and for all sufficiently large $r,$

[TABLE]

holds where $\delta>1.$

Lemma 2**.**

[5, p. 18]** Let $f$ be an entire function. Then for all sufficiently large values of $r,$

[TABLE]

3. Main Results

In this section we present some results which will be needed in the sequel.

Theorem 1**.**

Let $f_{1}$, $f_{2}$ be meromorphic functions and $g_{1}$ be any entire function such that at least $f_{1}$ or $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Also let $g_{1}$ has the Property (A). Then

[TABLE]

The equality holds when $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ where $i,j=1,2$ and $i\neq j$.

Proof.

The result is obvious when $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=0$. So we suppose that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)$ $>$ [math]. We can clearly assume that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{k},\varphi\right)$ is finite for $k=1,2.$ Now let us consider that $\max\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}=\Delta$ and $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}.$

Now for any arbitrary $\varepsilon>0$ from the definition of $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, we have for a sequence values of $r$ tending to infinity that

[TABLE]

[TABLE]

Also for any arbitrary $\varepsilon>0$ from the definition of $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\left(=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right)$, we obtain for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

Since $T_{f_{1}\pm f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)+O(1)$ for all large $r,$ so in view of $\left(\ref{50.1}\right)$ , $\left(\ref{9.1c}\right)$ and Lemma 2, we obtain for a sequence values of $r$ tending to infinity that

[TABLE]

[TABLE]

Therefore in view of Lemma 1 and Lemma 2, we obtain from $\left(\ref{9.2}\right)$ for a sequence values of $r$ tending to infinity and $\sigma>1$ that

[TABLE]

[TABLE]

[TABLE]

Now we get from above by letting $\sigma\rightarrow 1^{+}$

[TABLE]

Since $\varepsilon>0$ is arbitrary,

[TABLE]

Similarly, if we consider that $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ or both $f_{1}$ and $f_{2}$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1},$ then one can easily verify that

[TABLE]

Further without loss of any generality, let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and $f=f_{1}\pm f_{2}.$ Then in view of $\left(\ref{9.3}\right)$ we get that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $\leq$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ As, $f_{2}=\pm\left(f-f_{1}\right)$ and in this case we obtain that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $\leq$ $\max\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\}~{}.$ As we assume that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),$ therefore we have $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $\leq$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ and hence $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $=$ $\max\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}.$ Therefore, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)\mid i=1,2$ provided $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ Thus the theorem is established.

Theorem 2**.**

Let $f_{1}$ and $f_{2}$ be any two meromorphic functions and $g_{1}$ be an entire function such that such that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exists . Also let $g_{1}$ has the Property (A). Then

[TABLE]

The equality holds when $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

We omit the proof of Theorem 2 as it can easily be carried out in the line of Theorem 1.

Theorem 3**.**

Let $f_{1}$ be a meromorphic function and $g_{1}$, $g_{2}$ be any two entire functions such that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exists. Also let $g_{1}\pm g_{2}$ has the Property (A). Then

[TABLE]

The equality holds when $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Proof.

The result is obvious when $\lambda_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\infty$. So we suppose that $\lambda_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\infty$. We can clearly assume that $\lambda_{g_{k}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ is finite for $k=1,2.$ Further let $\Psi=\min\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\}.$ Now for any arbitrary $\varepsilon>0$ from the definition of $\lambda_{g_{k}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, we have for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

Since $T_{g_{1}\pm g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)+O(1)$ for all large $r,$, we obtain from above and Lemma 2 for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

Therefore in view of Lemma 1 and Lemma 2, we obtain from above for all sufficiently large values of $r$ and any $\sigma>1$ that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

As $\varepsilon>0$ is arbitrary, we get from above by letting $\sigma\rightarrow 1^{+}$

[TABLE]

Now without loss of any generality, we may consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $g=g_{1}\pm g_{2}.$ Then in view of $\left(\ref{99.3}\right)$ we get that $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Further, $g_{1}=\left(g\pm g_{2}\right)$ and in this case we obtain that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\geq$ $\min\left\{\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\}~{}.$ As we assume that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ therefore we have $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and hence $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\min\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\}.$ Therefore, $\lambda_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\mid i=1,2$ provided $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Thus the theorem follows.

Theorem 4**.**

Let $f_{1}$ be a meromorphic function and $g_{1}$, $g_{2}$ be any two entire functions such that $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}.$ If $g_{1}\pm g_{2}$ has the Property (A), then

[TABLE]

The equality holds when $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ where $i,j=1,2$ and $i\neq j.$

We omit the proof of Theorem 4 as it can easily be carried out in the line of Theorem 3.

Theorem 5**.**

Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $g_{1}\pm g_{2}$ has the Property (A). Then

[TABLE]

*when the following two conditions holds:

$\left(i\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$; and

$\left(ii\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$.

The equality holds when $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ holds simultaneously for $i,1,2;$ j=1,2\and $i\neq j.$*

Proof.

Let the conditions $\left(i\right)$ and $\left(ii\right)$ of the theorem hold. Therefore in view of Theorem 2 and Theorem 4 we get that

[TABLE]

Since $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ hold simultaneously for $i=1,2;$ j=1,2\and $i\neq j,$ we obtain that

[TABLE]

[TABLE]

Now in view of the conditions $\left(i\right)$ and $\left(ii\right)$ of the theorem, it follows from above that

[TABLE]

which is the condition for holding equality in $\left(\ref{9.xyz}\right)$.

Hence the theorem follows.

Theorem 6**.**

Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $g_{1},g_{2}$ and $g_{1}\pm g_{2}$ satisfy the Property (A). Then

[TABLE]

*when the following two conditions holds:

$\left(i\right)$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$; and

$\left(ii\right)$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$.

The equality holds when $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ hold simultaneously for $i=1,2;$ j=1,2\and $i\neq j.$*

Proof.

Suppose that the conditions $\left(i\right)$ and $\left(ii\right)$ of the theorem holds. Therefore in view of Theorem 1 and Theorem 3, we obtain that

[TABLE]

Since $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j$, we get that

[TABLE]

[TABLE]

Since condition $\left(i\right)$ and $\left(ii\right)$ of the theorem holds, it follows from above that

[TABLE]

which is the condition for holding equality in $\left(\ref{9.xya}\right)$.

Hence the theorem follows.

Theorem 7**.**

Let $f_{1}$, $f_{2}$ be any two meromorphic functions and $g_{1}$ be any entire function such that at least $f_{1}$ or $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Also let $g_{1}$ satisfy the Property (A). Then

[TABLE]

The equality holds when $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ where $i,j=1,2$ and $i\neq j$.

Proof.

Since $T_{f_{1}\cdot f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)$ for all large $r,$ therefore applying the same procedure as adopted in Theorem 1 we get that

[TABLE]

Now without loss of any generality, let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and $f=f_{1}\cdot f_{2}.$ Then $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)\leq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ Further, $f_{2}=\frac{f}{f_{1}}$ and $T_{f_{1}}\left(r\right)=T_{\frac{1}{f_{1}}}\left(r\right)+O(1).$ Therefore $T_{f_{2}}\left(r\right)\leq T_{f}\left(r\right)+T_{f_{1}}\left(r\right)+O(1)$ and in this case we obtain that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\leq\max\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\}~{}.$ As we assume that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),$ therefore we have $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\leq\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ and hence $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $=$ $\max$ $\{$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $\}.$ Therefore, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)\mid i=1,2$ provided $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$

Hence the theorem follows.

Next we prove the result for the quotient $\frac{f_{1}}{f_{2}},$ provided $\frac{f_{1}}{f_{2}}$ is meromorphic.

Theorem 8**.**

Let $f_{1}$, $f_{2}$ be any two meromorphic functions and $g_{1}$ be any entire function such that at least $f_{1}$ or $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Also let $g_{1}$ satisfy the Property (A). Then

[TABLE]

provided $\frac{f_{1}}{f_{2}}$ is meromorphic. The equality holds when at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ and $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

Proof.

Since $T_{{}_{f_{2}}}\left(r\right)=T_{{}_{\frac{1}{f_{2}}}}\left(r\right)+O(1)$ and $T_{{}_{\frac{f_{1}}{f_{2}}}}\left(r\right)\leq T_{{}_{f_{1}}}\left(r\right)+T_{{}_{\frac{1}{f_{2}}}}\left(r\right),$ we get in view of Theorem 1 that

[TABLE]

Now in order to prove the equality conditions, we discuss the following two cases:

**Case I. **Suppose $\frac{f_{1}}{f_{2}}\left(=h\right)$ satisfies the following condition

[TABLE]

and $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}.$

Now if possible, let $\lambda_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$. Therefore from $f_{1}=h\cdot f_{2}$ we get that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ which is a contradiction. Therefore $\lambda_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)\geq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and in view of $\left(\ref{50.3}\right)$, we get that

[TABLE]

**Case II. ** Suppose $\frac{f_{1}}{f_{2}}\left(=h\right)$ satisfies the following condition

[TABLE]

and $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}.$

Now from $f_{1}=h\cdot f_{2}$ we get that either $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\leq\lambda_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)$ or $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\leq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$. But according to our assumption $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\nleq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$. Therefore $\lambda_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)\geq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and in view of $\left(\ref{50.3}\right)$, we get that

[TABLE]

Hence the theorem follows.

Now we state the following theorem which can easily be carried out in the line of Theorem 7 and Theorem 8 and therefore its proof is omitted.

Theorem 9**.**

Let $f_{1}$ and $f_{2}$ be any two meromorphic functions and $g_{1}$ be any entire function such that such that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exists. Also let $g_{1}$ satisfy the Property (A). Then

[TABLE]

The equality holds when $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$. Similar results hold for the quotient $\frac{f_{1}}{f_{2}}$, provided $\frac{f_{1}}{f_{2}}$ is meromorphic.

Theorem 10**.**

Let $f_{1}$ be a meromorphic function and $g_{1}$, $g_{2}$ be any two entire functions such that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exists. Also let $g_{1}\cdot g_{2}$ satisfy the Property (A). Then

[TABLE]

The equality holds when $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ where $i,j=1,2$ and $i\neq j$ and $g_{i}$ satisfy the Property (A). Similar results hold for the quotient $\frac{g_{1}}{g_{2}}$, provided $\frac{g_{1}}{g_{2}}$ is entire and satisfy the Property (A). The equality holds when $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $g_{1}$ satisfy the Property (A).

Proof.

Since $T_{g_{1}\cdot g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)$ for all large $r,$ therefore applying the same procedure as adopted in Theorem 3 we get that

[TABLE]

Now without loss of any generality, we may consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $g=g_{1}\cdot g_{2}.$ Then $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Further, $g_{1}=\frac{g}{g_{2}}$ and and $T_{g_{2}}\left(r\right)=T_{\frac{1}{g_{2}}}\left(r\right)+O(1).$ Therefore $T_{g_{1}}\left(r\right)$ $\leq$ $T_{g}\left(r\right)$ $+$ $T_{g_{2}}\left(r\right)$ $+$ $O(1)$ and in this case we obtain that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\min\left\{\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\}~{}.$ As we assume that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ so we have $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and hence $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\min$ $\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\}.$ Therefore, $\lambda_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\mid i=1,2$ provided $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $g_{1}$ satisfy the Property (A). Hence the first part of the theorem follows.

Now we prove our results for the quotient $\frac{g_{1}}{g_{2}}$, provided $\frac{g_{1}}{g_{2}}$ is entire and $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Since $T_{{}_{g_{2}}}\left(r\right)=T_{{}_{\frac{1}{g_{2}}}}\left(r\right)+O(1)$ and $T_{{}_{\frac{g_{1}}{g_{2}}}}\left(r\right)\leq T_{{}_{g_{1}}}\left(r\right)+T_{{}_{\frac{1}{g_{2}}}}\left(r\right),$ we get in view of Theorem 3 that

[TABLE]

Now in order to prove the equality conditions, we discuss the following two cases:

**Case I. **Suppose $\frac{g_{1}}{g_{2}}\left(=h\right)$ satisfies the following condition

[TABLE]

Now if possible, let $\lambda_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore from $g_{1}=h\cdot g_{2}$ we get that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, which is a contradiction. Therefore $\lambda_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\leq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and in view of $\left(\ref{50.11}\right)$, we get that

[TABLE]

**Case II. ** Suppose that $\frac{g_{1}}{g_{2}}\left(=h\right)$ satisfies the following condition

[TABLE]

Therefore from $g_{1}=h\cdot g_{2}$, we get that either $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\lambda_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. But according to our assumption $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\ngeqslant\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore $\lambda_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\leq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and in view of $\left(\ref{50.11}\right)$, we get that

[TABLE]

Hence the theorem follows.

Theorem 11**.**

Let $f_{1}$ be any meromorphic function and $g_{1}$, $g_{2}$ be any two entire functions such that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exists. Further let $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}.$ Also let $g_{1}\cdot g_{2}$ satisfy the Property (A). Then

[TABLE]

The equality holds when $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ where $i,j=1,2$ and $i\neq j$ and $g_{i}$ satisfy the Property (A).

Theorem 12**.**

Let $f_{1}$ be any meromorphic function and $g_{1}$, $g_{2}$ be any two entire functions such that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exists. Further let $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}.$ Then

[TABLE]

provided $\frac{g_{1}}{g_{2}}$ is entire and satisfy the Property (A). The equality holds when at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$, $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $g_{1}$ satisfy the Property (A).

We omit the proof of Theorem 11 and Theorem 12 as those can easily be carried out in the line of Theorem 10.

Now we state the following four theorems without their proofs as those can easily be carried out in the line of Theorem 5 and Theorem 6 respectively.

Theorem 13**.**

Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $g_{1}\cdot g_{2}$ be satisfy the Property (A). Then

[TABLE]

*when the following two conditions holds:

$\left(i\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ and $g_{i}$ satisfy the Property (A) for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$; and

$\left(ii\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ and $g_{i}$ satisfy the Property (A) for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$.

The quality holds when $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j.$*

Theorem 14**.**

Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $g_{1}\cdot g_{2}$, $g_{1}$ and $g_{2}$ be satisfy the Property (A). Then

[TABLE]

*when the following two conditions holds:

$\left(i\right)$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$; and

$\left(ii\right)$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$.

The equality holds when $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j.$*

Theorem 15**.**

Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions such that $\frac{f_{1}}{f_{2}}$ is meromorphic and $\frac{g_{1}}{g_{2}}$ is entire. Also let $\frac{g_{1}}{g_{2}}$ satisfy the Property (A). Then

[TABLE]

*when the following two conditions holds:

$\left(i\right)$ At least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$; and

$\left(ii\right)$ At least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

The equality holds when $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j.$*

Theorem 16**.**

Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions such that $\frac{f_{1}}{f_{2}}$ is meromorphic and $\frac{g_{1}}{g_{2}}$ is entire. Also let $\frac{g_{1}}{g_{2}}$, $g_{1}$ and $g_{2}$ be satisfy the Property (A). Then

[TABLE]

*when the following two conditions hold:

$\left(i\right)$ At least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ and $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$; and

$\left(ii\right)$ At least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

The equality holds when $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j.$*

Next we intend to find out the sum and product theorems of relative $\left(p,q\right)$-$\varphi$ type ( respectively relative $\left(p,q\right)$-$\varphi$ lower type) and relative $\left(p,q\right)$-$\varphi$ weak type of meromorphic function with respect to an entire function taking into consideration of the above theorems.

Theorem 17**.**

*Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ are all non zero and finite.

**(A) *If $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ for $i,$ $j$ $=$ $1,2$; $i\neq j,$ and $g_{1}$ has the Property (A), then

[TABLE]

(B) If $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$, $j$ $=$ $1,2$; $i\neq j$ and $g_{1}\pm g_{2}$ has the Property (A), then

[TABLE]

**(C) Assume the functions $f_{1},f_{2},g_{1}$ and $g_{2}$ satisfy the following conditions:

$\left(i\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(ii\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(iii\right)$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j$;

$\left(iv\right)\rho_{g_{m}}^{\left(p,q\right)}\left(f_{l},\varphi\right)=\max\left[\min\left\{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\},\min\left\{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}\right]\mid l,m=1,2$, and $g_{1}\pm g_{2}$ has the Property (A);

then**

[TABLE]

and

[TABLE]

Proof.

From the definition of relative $\left(p,q\right)$-$\varphi$ type and relative $\left(p,q\right)$-$\varphi$ lower type of meromorphic function with respect to an entire function, we have for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

and for a sequence of values of $r$ tending to infinity, we obtain that

[TABLE]

and

[TABLE]

where $\varepsilon>0$ is any arbitrary positive number $k=1,\,2$ and $l=1,2.$

Case I. Suppose that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ hold. Also let $\varepsilon\left(>0\right)$ be arbitrary. Since $T_{f_{1}\pm f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)+O(1)$ for all large $r,$ so in view of $\left(\ref{9.15}\right),$ we get for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

where $A=\frac{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)}\right\}\right]+O(1)}{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]},$ and in view of $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, and for all sufficiently large values of $r$, we can make the term $A$ sufficiently small . Hence for any $\alpha=1+\varepsilon_{1}$, it follows from $\left(\ref{9.18}\right)$ for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

Hence making $\alpha\rightarrow 1+,$ we get in view of Theorem 2, $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and above for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

Now we may consider that $f=f_{1}\pm f_{2}.$ Since $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ hold. Then $\sigma_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\leq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Further, let $f_{1}=\left(f\pm f_{2}\right)$. Therefore in view of Theorem 2 and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, we obtain that $\rho_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds. Hence in view of $\left(\ref{9.21}\right)$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\leq\sigma_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right).$ Therefore $\sigma_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\Rightarrow$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Similarly, if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),$ then one can easily verify that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

Case II. Let us consider that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ hold. Also let $\varepsilon\left(>0\right)$ are arbitrary. Since $T_{f_{1}\pm f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)+O(1)$ for all large $r,$ from $\left(\ref{9.15}\right)$ and $\left(\ref{9.20a}\right),$ we get for a sequence of values of $r$ tending to infinity that

[TABLE]

[TABLE]

where $B=\frac{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)}\right\}\right]+O(1)}{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]},$ and in view of $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, we can make the term $B$ sufficiently small by taking $n$ sufficiently large and therefore using the similar technique for as executed in the proof of Case I we get from $\left(\ref{9.185}\right)$ that $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ when $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ hold. Likewise, if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),$ then one can easily verify that $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

Thus combining Case I and Case II, we obtain the first part of the theorem.

Case III. Let us consider that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}.$ We can make the term $C=\frac{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]+O(1)}{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]}$ sufficiently small by taking $n$ sufficiently large, since $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Hence $C<\varepsilon_{1}.$

As $T_{g_{1}\pm g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)+O(1)$ for all large $r$, we get that

[TABLE]

[TABLE]

[TABLE]

Therefore for any $\alpha=1+\varepsilon_{1},$ we obtain in view of $C<\varepsilon_{1},$ $\left(\ref{9.15a}\right)$ and $\left(\ref{9.20}\right)$ for a sequence of values of $r$ tending to infinity that

[TABLE]

Now making $\alpha\rightarrow 1+$, we obtain from above for a sequence of values of $r$ tending to infinity that

[TABLE]

Since $\varepsilon>0$ is arbitrary, we find that

[TABLE]

Now we may consider that $g=g_{1}\pm g_{2}.$ Also $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$. Then $\sigma_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Further let $g_{1}=\left(g\pm g_{2}\right)$. Therefore in view of Theorem 4 and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, we obtain that $\rho_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ as at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$. Hence in view of $\left(\ref{9.237}\right)$, $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\sigma_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Therefore $\sigma_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\Rightarrow$ $\sigma_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Similarly if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$, then $\sigma_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$

**Case IV. **In this case suppose that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}.$ we can also make the term $D=\frac{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]+O(1)}{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]}$ sufficiently small by taking $r$ sufficiently large as $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ So $D<\varepsilon_{1}$ for sufficiently large $r.$ As $T_{g_{1}\pm g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)+O(1)$ for all large $r$, therefore from $\left(\ref{9.15a}\right),$ we get for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and therefore using the similar technique for as executed in the proof of Case III we get from $\left(\ref{9.238}\right)$ that $\overline{\sigma}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ where $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$.

Likewise if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$, then $\overline{\sigma}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$

Thus combining Case III and Case IV, we obtain the second part of the theorem.

The third part of the theorem is a natural consequence of Theorem 5 and the first part and second part of the theorem. Hence its proof is omitted.

Theorem 18**.**

*Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ are all non zero and finite.

(A) If $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ for $i$, $j$ $=$ $1,2$; $i\neq j$, and $g_{1}$ has the Property (A), then*

[TABLE]

(B) If $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ for $i$, $j$ $=$ $1,2$; $i\neq j$ and $g_{1}\pm g_{2}$ has the Property (A), then

[TABLE]

**(C) Assume the functions $f_{1},f_{2},g_{1}$ and $g_{2}$ satisfy the following conditions:

$\left(i\right)$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ for $i$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(ii\right)$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ for $i$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(iii\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds simultaneously for $i,$ j=1,2\and $i\neq j$;

$\left(iv\right)\lambda_{g_{m}}^{\left(p,q\right)}\left(f_{l},\varphi\right)=\min\left[\max\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\},\max\left\{\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}\right]\mid l,m=1,2$ and $g_{1}\pm g_{2}$ has the Property (A)

then we have**

[TABLE]

and

[TABLE]

Proof.

For any arbitrary positive number $\varepsilon(>0)$, we have for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

and for a sequence of values of $r$ tending to infinity we obtain that

[TABLE]

and

[TABLE]

where $k=1,2$ and $l=1,2.$

Case I. Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Also let $\varepsilon\left(>0\right)$ be arbitrary. Since $T_{f_{1}\pm f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)+O(1)$ for all large $r,$ we get from $\left(\ref{9.15x}\right)$ and $\left(\ref{9.20ax}\right),$ for a sequence $\left\{r_{n}\right\}$ of values of $r$ tending to infinity that

[TABLE]

[TABLE]

where $E=\frac{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)}\right\}\right]+O(1)}{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]}$ and in view of $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2}\right)$, we can make the term $E$ sufficiently small by taking $n$ sufficiently large. Now with the help of Theorem 1 and using the similar technique of Case I of Theorem 17, we get from $\left(\ref{9.18x}\right)$ that

[TABLE]

Further, we may consider that $f=f_{1}\pm f_{2}.$ Also suppose that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Then $\tau_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\leq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Now let $f_{1}=\left(f\pm f_{2}\right)$. Therefore in view of Theorem 1, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1},$ we obtain that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds. Hence in view of $\left(\ref{9.21x}\right)$, $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\leq\tau_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right).$ Therefore $\tau_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\Rightarrow$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Similarly, if we consider $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ then one can easily verify that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

Case II. Let us consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Also let $\varepsilon\left(>0\right)$ be arbitrary. As $T_{f_{1}\pm f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)+O(1)$ for all large $r,$ we obtain from $\left(\ref{9.15x}\right)$ for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

where $F=\frac{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)}\right\}\right]+O(1)}{T_{g_{1}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)+\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1,\varphi}\right)}\right\}\right]},$ and in view of $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, we can make the term $F$ sufficiently small by taking $r$ sufficiently large and therefore for similar reasoning of Case I we get from $\left(\ref{9.185x}\right)$ that $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ when $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$.

Likewise, if we consider $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ then one can easily verify that $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)=\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$

Thus combining Case I and Case II, we obtain the first part of the theorem.

Case III. Let us consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore we can make the term $G=\frac{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]+O(1)}{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r\right)\right]^{\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]}$ sufficiently small by taking $r$ sufficiently large since $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ So $G<\varepsilon_{1}.$ Since $T_{g_{1}\pm g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)+O(1)$ for all large $r,$ we get from $\left(\ref{9.15ax}\right)$ for all sufficiently large values of $r$ that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Therefore in view of Theorem 3 and using the similar technique of Case III of Theorem 17, we get from $\left(\ref{9.236x}\right)$ that

[TABLE]

Further, we may consider that $g=g_{1}\pm g_{2}.$ As $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, so $\tau_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Further let $g_{1}=\left(g\pm g_{2}\right)$. Therefore in view of Theorem 3 and $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ we obtain that $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ holds. Hence in view of $\left(\ref{9.237x}\right)$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\tau_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Therefore $\tau_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\Rightarrow$ $\tau_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Likewise, if we consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ then one can easily verify that $\tau_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$

**Case IV. **In this case further we consider $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Further we can make the term $H=\frac{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]+O(1)}{T_{g_{2}}\left[\exp^{\left[p-1\right]}\left\{\left(\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)-\varepsilon\right)\left[\log^{\left[q-1\right]}\varphi\left(r_{n}\right)\right]^{\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)}\right\}\right]}$ sufficiently small by taking $n$ sufficiently large, since $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Therefore $H<\varepsilon_{1}$ for sufficiently large $n.$ As $T_{g_{1}\pm g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)+O(1)$ for all large $r,$ we obtain from $\left(\ref{9.15ax}\right)$ and $\left(\ref{9.20x}\right),$ we obtain for a sequence $\left\{r_{n}\right\}$ of values of $r$ tending to infinity that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and therefore using the similar technique for as executed in the proof of Case IV of Theorem 17, we get from $\left(\ref{9.238x}\right)$ that $\overline{\tau}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ when $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Similarly, if we consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ then one can easily verify that $\overline{\tau}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$

Thus combining Case III and Case IV, we obtain the second part of the theorem.

The proof of the third part of the Theorem is omitted as it can be carried out in view of Theorem 6 and the above cases.

In the next two theorems we reconsider the equalities in Theorem 1 to Theorem 4 under somewhat different conditions.

Theorem 19**.**

*Let $f_{1},\,f_{2}\,$be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions.

**(A) **The following condition is assumed to be satisfied:

$\left(i\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds and $g_{1}$ has the Property (A), then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ holds and $g_{1}\pm g_{2}$ has the Property (A);

$\left(ii\right)$ $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}$, then**

[TABLE]

Proof.

Let $f_{1},\,f_{2},\,g_{1}$ and $g_{2}$ be any four entire functions satisfying the conditions of the theorem.

Case I. Suppose that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $(0<$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $<\infty)$. Now in view of Theorem 2 it is easy to see that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\leq\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)~{}.$ If possible let

[TABLE]

Let** $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$** Then in view of the first part of Theorem 17 and $\left(\ref{93.1x}\right)$ we obtain that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2}\mp f_{2},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ which is a contradiction. Hence $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)~{}.$ Similarly with the help of the first part of Theorem 17, one can obtain the same conclusion under the hypothesis $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ This proves the first part of the theorem.

**Case II. **Let us consider that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $(0<$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<\infty)$, $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}$ and $\left(g_{1}\pm g_{2}\right)$ and $g_{1}\pm g_{2}$ satisfy the Property (A). Therefore in view of Theorem 4, it follows that $\rho_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and if possible let

[TABLE]

Let us consider that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Then. in view of the proof of the second part of Theorem 17 and $\left(\ref{93.3x}\right)$ we obtain that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{1}\pm g_{2}\mp g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ which is a contradiction. Hence $\rho_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)~{}.$ Also in view of the proof of second part of Theorem 17 one can derive the same conclusion for the condition $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and therefore the second part of the theorem is established.

Theorem 20**.**

*Let $f_{1},\,f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions.

**(A) **The following conditions are assumed to be satisfied:

$\left(i\right)$ $\left(f_{1}\pm f_{2}\right)$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}$, and $g_{1},$ $g_{2}$ , $g_{1}\pm g_{2}$ have the Property (A);

$\left(ii\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)$;

$\left(iii\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(iv\right)$ Either $\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$; then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ $f_{1}$ and $f_{2}$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2},$ and $g_{1}\pm g_{2}$ has the Property (A);

$\left(ii\right)$ Either $\sigma_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(iii\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$;

$\left(iv\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$; then**

[TABLE]

We omit the proof of Theorem 20 as it is a natural consequence of Theorem 19.

Theorem 21**.**

*Let $f_{1},\,f_{2}\,$be ant two meromorphic functions and $g_{1}$,$g_{2}$ be any two entire functions.

(A) The following conditions are assumed to be satisfied:

$\left(i\right)$ At least any one of $f_{1}$ or $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$;

$\left(ii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds and $g_{1}$ has the Property (A), then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ $f_{1},$ $g_{1}$ and $g_{2}$ be any three entire functions such that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exists;

$\left(ii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ holds and $g_{1}\pm g_{2}$ has the Property (A), then**

[TABLE]

Proof.

Let $f_{1},\,f_{2},\,g_{1}$ and $g_{2}$ be any four entire functions satisfying the conditions of the theorem.

Case I. Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $(0<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\infty)$ and at least $f_{1}$ or $f_{2}$ and $\left(f_{1}\pm f_{2}\right)$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Now, in view of Theorem 1, it is easy to see that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\leq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ If possible let

[TABLE]

Let** $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$** Then in view of the proof of the first part of Theorem 18 and $\left(\ref{96.19xy}\right)$ we obtain that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2}\mp f_{2},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ which is a contradiction. Hence $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)~{}.$ Similarly in view of the proof of the first part of Theorem 18 , one can establish the same conclusion under the hypothesis $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ This proves the first part of the theorem.

Case II. Let us consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $(0<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\infty.$ Therefore in view of Theorem 3, it follows that $\lambda_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and if possible let

[TABLE]

Suppose** $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$** Then in view of the second part of Theorem 18 and $\left(\ref{96.2t}\right)$, we obtain that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{1}\pm g_{2}\mp g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ which is a contradiction. Hence $\lambda_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)~{}.$ Analogously with the help of the second part of Theorem 18, the same conclusion can also be derived under the condition $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and therefore the second part of the theorem is established.

Theorem 22**.**

*Let $f_{1},\,f_{2}\,$be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions.

(A) The following conditions are assumed to be satisfied:

$\left(i\right)$ At least any one of $f_{1}$ or $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ and $g_{2}$. Also $g_{1},$ $g_{2},$ $g_{1}\pm g_{2}$ have satisfy the Property (A);

$\left(ii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1}\pm f_{2},\varphi\right)$;

$\left(iii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(iv\right)$ Either $\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$; then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ At least any one of $f_{1}$ or $f_{2}$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}\pm g_{2}$, and $g_{1}\pm g_{2}$ has satisfy the Property (A);

$\left(ii\right)$ Either $\tau_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}\pm g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds;

$\left(iii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ holds;

$\left(iv\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds, then**

[TABLE]

We omit the proof of Theorem 22 as it is a natural consequence of Theorem 21.

Theorem 23**.**

*Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ are all non zero and finite.

(A) Assume the functions $f_{1},f_{2}$ and $g_{1}$ satisfy the following conditions:

$\left(i\right)$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ for $i$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(ii\right)$ $g_{1}$ satisfies the Property (A), then*

[TABLE]

Similarly,

[TABLE]

*holds provided $\left(i\right)$ $\frac{f_{1}}{f_{2}}$ is meromorphic, $\left(ii\right)$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)$ $>$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ $\mid$ $i$, $1,2;$ $j$ $=$ $1,2;$ $i$ $\neq$ $j$ and $\left(iii\right)$ $g_{1}$ satisfy the Property (A).

**(B) **Assume the functions $g_{1},g_{2}$ and $f_{1}$ satisfy the following conditions:

$\left(i\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$, $j$ $=$ $1,2$ and $i\neq j,$ and $g_{i}$ satisfy the Property (A);

$\left(ii\right)$ $g_{1}\cdot g_{2}$ satisfy the Property (A), then*

[TABLE]

Similarly,

[TABLE]

*holds provided $\left(i\right)$ $\frac{g_{1}}{g_{2}}$ is entire and satisfy the Property (A), $\left(ii\right)$ At least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$, $\left(iii\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\mid$ $i$ $=$ $1,2;$ $j$ $=$ $1,2;$ $i$ $\neq$ $j$ and $\left(iv\right)$ $g_{1}$ satisfy the Property (A).

(C) Assume the functions $f_{1},f_{2}$, $g_{1}$ and $g_{2}$ satisfy the following conditions:

$\left(i\right)$ $g_{1}\cdot g_{2}$ satisfy the Property (A);

$\left(ii\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(iii\right)$ $\rho_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\rho_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{j}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(iv\right)$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j$;

$\left(v\right)$ $\rho_{g_{m}}^{\left(p,q\right)}\left(f_{l},\varphi\right)=\max\left[\min\left\{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\},\min\left\{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}\right]\mid l,m=1,2$; then*

[TABLE]

Similarly,

[TABLE]

*holds provided $\frac{f_{1}}{f_{2}}$ is meromorphic function and $\frac{g_{1}}{g_{2}}$ is entire function which satisfy the following conditions:

$\left(i\right)$ $\frac{g_{1}}{g_{2}}$ satisfy the Property (A);

$\left(ii\right)$ At least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$;

$\left(iii\right)$ At least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(iv\right)$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j$;

$\left(v\right)$ $\rho_{g_{m}}^{\left(p,q\right)}\left(f_{l},\varphi\right)=\max\left[\min\left\{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\right\},\min\left\{\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}\right]\mid l,m=1,2$.*

Proof.

Let us suppose that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ are all non zero and finite.

Case I. Suppose that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$. Also let $g_{1}$ satisfy the Property (A).  Since $T_{f_{1}\cdot f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)$ for all large $r,$ therefore applying the same procedure as adopted in Case I of Theorem 17 we get that

[TABLE]

Further without loss of any generality, let $f=f_{1}\cdot f_{2}$ and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $<$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right).$ Then in view of $\left(\ref{77.3}\right),$ we obtain that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $\leq$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Also $f_{1}=\frac{f}{f_{2}}$ and $T_{f_{2}}\left(r\right)$ $=$ $T_{\frac{1}{f_{2}}}\left(r\right)$ $+$ $O(1).$ Therefore $T_{f_{1}}\left(r\right)\leq T_{f}\left(r\right)+T_{f_{2}}\left(r\right)+O(1)$ and in this case also we obtain from $\left(\ref{77.3}\right)$ that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\leq$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right).$ Hence $\sigma_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\Rightarrow$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$

Similarly, if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),$ then one can verify that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$

Next we may suppose that $f=\frac{f_{1}}{f_{2}}$ with $f_{1},$ $f_{2}$ and $f$ are all meromorphic functions.

Sub Case IA. Let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $<$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 9, $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $<$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$. We have $f_{1}=f\cdot f_{2}$. So, $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)$.

**Sub Case IB. **Let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $>$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 9, $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$. Since $T_{f}\left(r\right)=T_{\frac{1}{f}}\left(r\right)+O(1)=T_{\frac{f_{2}}{f_{1}}}\left(r\right)+O(1),$ So $\sigma_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

**Case II. **Let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$. Also let $g_{1}$ satisfy the Property (A). As $T_{f_{1}\cdot f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)$ for all large $r,$ therefore applying the same procedure as explored in Case II of Theorem 17, one can easily verify that $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)=\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)$ $=$ $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)\mid i=1,2$  under the conditions specified in the theorem.

Similarly, if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right),$ then one can verify that $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ and $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)$ $=$ $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$

Therefore the first part of theorem follows from Case I and Case II.

**Case III. **Let $g_{1}\cdot g_{2}$ satisfy the Property (A) and $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}.$ Since $T_{g_{1}\cdot g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)$ for all large $r,$ therefore applying the same procedure as adopted in Case III of Theorem 17 we get that

[TABLE]

Further without loss of any generality, let $g=g_{1}\cdot g_{2}$ and $\rho_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Then in view of $\left(\ref{7.6j}\right),$ we obtain that $\sigma_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\geq$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Also $g_{1}=\frac{g}{g_{2}}$ and $T_{g_{2}}\left(r\right)$ $=$ $T_{\frac{1}{g_{2}}}\left(r\right)$ $+$ $O(1).$ Therefore $T_{g_{1}}\left(r\right)\leq T_{g}\left(r\right)+T_{g_{2}}\left(r\right)+O(1)$ and in this case we obtain from $\left(\ref{7.6j}\right)$ that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\geq$  $\sigma_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Hence $\sigma_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\Rightarrow$ $\sigma_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Similarly, if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$, then one can verify that $\sigma_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Next we may suppose that $g=\frac{g_{1}}{g_{2}}$ with $g_{1},$ $g_{2},$ $g$ are all entire functions satisfying the conditions specified in the theorem.

Sub Case IIIA. Let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 12, $\rho_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. We have $g_{1}=g\cdot g_{2}$. So $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=\sigma_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

**Sub Case IIIB. **Let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $>$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 12, $\rho_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Since $T_{g}\left(r\right)=T_{\frac{1}{g}}\left(r\right)+O(1)=T_{\frac{g_{2}}{g_{1}}}\left(r\right)+O(1),$ So $\sigma_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

**Case IV. **Suppose $g_{1}\cdot g_{2}$ satisfy the Property (A). Also let $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}.$ As $T_{g_{1}\cdot g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)$ for all large $r,$ the same procedure as explored in Case IV of Theorem 17, one can easily verify that $\overline{\sigma}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\overline{\sigma}_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\overline{\sigma}_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\mid i=1,2$ under the conditions specified in the theorem.

Likewise, if we consider $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$, then one can verify that $\overline{\sigma}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\overline{\sigma}_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore the second part of theorem follows from Case III and Case IV.

Proof of the third part of the Theorem is omitted as it can be carried out in view of Theorem 13 and Theorem 15 and the above cases.

Theorem 24**.**

*Let $f_{1},f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions. Also let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ are all non zero and finite.

(A) Assume the functions $f_{1},f_{2}$ and $g_{1}$ satisfy the following conditions:

$\left(i\right)$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ for $i$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(ii\right)$ $g_{1}$ satisfy the Property (A), then*

[TABLE]

Similarly,

[TABLE]

*holds provided $\frac{f_{1}}{f_{2}}$ is meromorphic, at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ where $g_{1}$ satisfy the Property (A) and $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)$ $>$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ $\mid$ $i$ $=$ $1,2;$ $j$ $=$ $1,2;$ $i$ $\neq$ $j$.

**(B) **Assume the functions $g_{1},g_{2}$ and $f_{1}$ satisfy the following conditions:

$\left(i\right)$ $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ for $i$, $j$ $=$ $1,2$, $i\neq j$; and $g_{i}$ satisfy the Property (A)

$\left(ii\right)$ $g_{1}\cdot g_{2}$ satisfy the Property (A), then*

[TABLE]

Similarly,

[TABLE]

*holds provided $\frac{g_{1}}{g_{2}}$ is entire and satisfy the Property (A), $g_{1}$ satisfy the Property (A) and $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\mid$ $i$ $=$ $1,2;$ $j$ $=$ $1,2;$ $i$ $\neq$ $j$.

(C) Assume the functions $f_{1},f_{2}$, $g_{1}$ and $g_{2}$ satisfy the following conditions:

$\left(i\right)$ $g_{1}\cdot g_{2}$, $g_{1}$ and $g_{2}$ are satisfy the Property (A);

$\left(ii\right)$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(iii\right)$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{i},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{j},\varphi\right)$ with at least $f_{j}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ for $i$ $=$ $1,$ $2$, $j$ $=$ $1,2$ and $i\neq j$;

$\left(iv\right)$ $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j$;

$\left(v\right)$ $\lambda_{g_{m}}^{\left(p,q\right)}\left(f_{l},\varphi\right)=\min\left[\max\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\},\max\left\{\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}\right]\mid l,m=1,2$; then*

[TABLE]

Similarly,

[TABLE]

*holds provided $\frac{f_{1}}{f_{2}}$ is meromorphic and $\frac{g_{1}}{g_{2}}$ is entire functions which satisfy the following conditions:

$\left(i\right)$ $\frac{g_{1}}{g_{2}}$, $g_{1}$ and $g_{2}$ satisfy the Property (A);

$\left(ii\right)$ At least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ and $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(iii\right)$ At least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{2}$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(iv\right)$ $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{i}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\lambda_{g_{j}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds simultaneously for $i=1,2;$ j=1,2\and $i\neq j$;

$\left(v\right)$ $\lambda_{g_{m}}^{\left(p,q\right)}\left(f_{l},\varphi\right)=\min\left[\max\left\{\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\},\max\left\{\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\right\}\right]\mid l,m=1,2$.*

Proof.

Let us consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$, $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ are all non zero and finite.

Case I. Suppose $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ and $g_{1}$ satisfy the Property (A). Since $T_{f_{1}\cdot f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)$ for all large $r,$ therefore applying the same procedure as adopted in Case I of Theorem 18 we get that

[TABLE]

Further without loss of any generality, let $f=f_{1}\cdot f_{2}$ and $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $<$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right).$ Then in view of $\left(\ref{77.90}\right),$ we obtain that $\tau_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $\leq$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Also $f_{1}=\frac{f}{f_{2}}$ and $T_{f_{2}}\left(r\right)$ $=$ $T_{\frac{1}{f_{2}}}\left(r\right)$ $+$ $O(1).$ Therefore $T_{f_{1}}\left(r\right)\leq T_{f}\left(r\right)+T_{f_{2}}\left(r\right)+O(1)$ and in this case we obtain from the above arguments that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\leq$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right).$ Hence $\tau_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\Rightarrow$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$

Similarly, if we consider $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$, then one can easily verify that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$

Next we may suppose that $f=\frac{f_{1}}{f_{2}}$ with $f_{1},$ $f_{2}$ and $f$ are all meromorphic functions satisfying the conditions specified in the theorem.

Sub Case IA. Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $<\lambda$ ${}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 8, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $<$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$. We have $f_{1}=f\cdot f_{2}$. So $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)$.

**Sub Case IB. **Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $>$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 8, $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f,\varphi\right)$. Since $T_{f}\left(r\right)=T_{\frac{1}{f}}\left(r\right)+O(1)=T_{\frac{f_{2}}{f_{1}}}\left(r\right)+O(1),$ So $\tau_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}}{f_{2}},\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$.

**Case II. **Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ where $g_{1}$ satisfy the Property (A). As $T_{f_{1}\cdot f_{2}}\left(r\right)\leq T_{f_{1}}\left(r\right)+T_{f_{2}}\left(r\right)$ for all large $r,$ so applying the same procedure as adopted in Case II of Theorem 18 we can easily verify that $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)=\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\overline{\tau}_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\overline{\tau}_{gi}^{\left(p,q\right)}\left(f_{1},\varphi\right)\mid i=1,2$ under the conditions specified in the theorem.

Similarly, if we consider $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ with at least $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$, then one can easily verify that $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$

Therefore the first part of theorem follows Case I and Case II.

**Case III. **Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $g_{1}\cdot g_{2}$ satisfy the Property (A).Since $T_{g_{1}\cdot g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)$ for all large $r,$ therefore applying the same procedure as adopted in Case III of Theorem 18 we get that

[TABLE]

Further without loss of any generality, let $g=g_{1}\cdot g_{2}$ and $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Then in view of $\left(\ref{79.0}\right),$ we obtain that $\tau_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\geq$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Also $g_{1}=\frac{g}{g_{2}}$ and $T_{g_{2}}\left(r\right)$ $=$ $T_{\frac{1}{g_{2}}}\left(r\right)$ $+$ $O(1).$ Therefore $T_{g_{1}}\left(r\right)\leq T_{g}\left(r\right)+T_{g_{2}}\left(r\right)+O(1)$ and in this case we obtain from above arguments that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\geq$  $\tau_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Hence $\tau_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\Rightarrow$ $\tau_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

If $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ then one can easily verify that $\tau_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Next we may suppose that $g=\frac{g_{1}}{g_{2}}$ with $g_{1},$ $g_{2},$ $g$ are all entire functions satisfying the conditions specified in the theorem.

Sub Case IIIA. Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 10, $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. We have $g_{1}=g\cdot g_{2}$. So $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=\tau_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

**Sub Case IIIB. **Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $>$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Therefore in view of Theorem 10, $\lambda_{g}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $<$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$. Since $T_{g}\left(r\right)=T_{\frac{1}{g}}\left(r\right)+O(1)=T_{\frac{g_{2}}{g_{1}}}\left(r\right)+O(1),$ So $\tau_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

**Case IV. **Suppose $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $g_{1}\cdot g_{2}$ satisfy the Property (A). Since $T_{g_{1}\cdot g_{2}}\left(r\right)\leq T_{g_{1}}\left(r\right)+T_{g_{2}}\left(r\right)$ for all large $r,$ then adopting the same procedure as of Case IV of Theorem 18, we obtain that $\overline{\tau}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\overline{\tau}_{\frac{g_{1}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\overline{\tau}_{gi}^{\left(p,q\right)}\left(f_{1},\varphi\right)\mid i=1,2$.

Similarly if we consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)>\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right),$ then one can easily verify that $\overline{\tau}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$.

Therefore the second part of the theorem follows from Case III and Case IV.

Proof of the third part of the Theorem is omitted as it can be carried out in view of Theorem 14 , Theorem 16 and the above cases.

Theorem 25**.**

*Let $f_{1},\,f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions.

(A) The following condition is assumed to be satisfied:

$\left(i\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds;

$\left(ii\right)$ $g_{1}$ satisfies the Property (A), then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ holds;

$\left(ii\right)$ $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}$. Also $g_{1}\cdot g_{2}$ satisfy the Property (A). Then we have**

[TABLE]

Proof.

Let $f_{1},\,f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions satisfying the conditions of the theorem.

Case I. Suppose that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $(0<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\infty)$ and $g_{1}$ satisfy the Property (A). Now in view of Theorem 9, it is easy to see that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)\leq\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ If possible let

[TABLE]

Let** $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$** Now in view of the first part of Theorem 23 and $\left(\ref{20.1}\right)$ we obtain that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}\cdot f_{2}}{f_{2}},\varphi\right)=\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ which is a contradiction. Hence $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ Similarly with the help of the first part of Theorem 23, one can obtain the same conclusion under the hypothesis $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ This prove the first part of the theorem.

Case II. Let us consider that $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $(0<\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\infty)$, $f_{1}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one of $g_{1}$ or $g_{2}$. Also $g_{1}\cdot g_{2}$ satisfy the Property (A). Therefore in view of Theorem 11, it follows that $\rho_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\geq\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and if possible let

[TABLE]

Further suppose that** $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$** Therefore in view of the proof of the second part of Theorem 23 and $\left(\ref{20.2}\right)$, we obtain that $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{\frac{g_{1}\cdot g_{2}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ which is a contradiction. Hence $\rho_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\rho_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)~{}.$ Likewise in view of the proof of second part of Theorem 23, one can obtain the same conclusion under the hypothesis $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ This proves the second part of the theorem.

Theorem 26**.**

*Let $f_{1},\,f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions.

**(A) **The following conditions are assumed to be satisfied:

$\left(i\right)$ $\left(f_{1}\cdot f_{2}\right)$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one $g_{1}$ or $g_{2}$;

$\left(ii\right)$ $\left(g_{1}\cdot g_{2}\right)$, $g_{1}$ and $g_{2}$ all satisfy the Property (A);

$\left(iii\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$;

$\left(iv\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(v\right)$ Either $\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$; then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ $\left(g_{1}\cdot g_{2}\right)$ satisfy the Property (A);

$\left(ii\right)$ $f_{1}$ and $f_{2}$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to at least any one $g_{1}$ or $g_{2}$;

$\left(iii\right)$ Either $\sigma_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(iv\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$;

$\left(v\right)$ Either $\sigma_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\sigma_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\sigma}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\overline{\sigma}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$; then**

[TABLE]

We omit the proof of Theorem 26 as it is a natural consequence of Theorem 25.

Theorem 27**.**

*Let $f_{1},\,f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions.

(A) The following conditions are assumed to be satisfied:

$\left(i\right)$ At least any one of $f_{1}$ or $f_{2}$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$;

$\left(ii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds.

$\left(iii\right)$ $g_{1}$ satisfy the Property (A), then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ $f_{1}$ be any meromorphic function and $g_{1}$, $g_{2}$ be any two entire functions such that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ exist and $g_{1}\cdot g_{2}$ satisfy the Property (A);

$\left(ii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ holds, then**

[TABLE]

Proof.

Let $f_{1},\,f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions satisfy the conditions of the theorem.

Case I. Let $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ $(0<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)<\infty),$ $g_{1}$ satisfy the Property (A) and at least $f_{1}$ or $f_{2}$ is of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$. Now in view of Theorem 7 it is easy to see that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $\leq$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ If possible let

[TABLE]

Also let** $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$** Then in view of the proof of first part of Theorem 24 and $\left(\ref{20.3}\right),$ we obtain that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(\frac{f_{1}\cdot f_{2}}{f_{2}},\varphi\right)=\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ which is a contradiction. Hence $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right).$ Analogously, in view of the proof of first part of Theorem 24, one can derived the same conclusion under the hypothesis $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$. Hence the first part of the theorem is established.

Case II. Let us consider that $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $(0<\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right),\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)<\infty$ and $g_{1}\cdot g_{2}$ satisfy the Property (A). Therefore in view of Theorem 10, it follows that $\lambda_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $\geq$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and if possible let

[TABLE]

Further let** $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$** Then in view of second part of Theorem 24 and $\left(\ref{20.4}\right)$, we obtain that $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{\frac{g_{1}\cdot g_{2}}{g_{2}}}^{\left(p,q\right)}\left(f_{1},\varphi\right)=\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ which is a contradiction. Hence $\lambda_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ $=$ $\lambda_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right).$ Similarly by second part of Theorem 24, we get the same conclusion when $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ and therefore the second part of the theorem follows.

Theorem 28**.**

*Let $f_{1},\,f_{2}$ be any two meromorphic functions and $g_{1}$, $g_{2}$ be any two entire functions.

(A) The following conditions are assumed to be satisfied:

$\left(i\right)$ $g_{1}\cdot g_{2}$, $g_{1}$ and $g_{2}$ satisfy the Property (A);

$\left(ii\right)$ At least any one of $f_{1}$ or $f_{2}$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}$ and $g_{2}$;

$\left(iii\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1}\cdot f_{2},\varphi\right)$;

$\left(iv\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$;

$\left(v\right)$ Either $\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$; then*

[TABLE]

**(B) The following conditions are assumed to be satisfied:

$\left(i\right)$ $g_{1}\cdot g_{2}$ satisfy the Property (A);

$\left(ii\right)$ At least any one of $f_{1}$ or $f_{2}$ are of regular relative $\left(p,q\right)$-$\varphi$ growth with respect to $g_{1}\cdot g_{2}$;

$\left(iii\right)$ Either $\tau_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{1}\cdot g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds;

$\left(iv\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{1},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{1},\varphi\right)$ holds;

$\left(v\right)$ Either $\tau_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\tau_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ or $\overline{\tau}_{g_{1}}^{\left(p,q\right)}\left(f_{2},\varphi\right)\neq\overline{\tau}_{g_{2}}^{\left(p,q\right)}\left(f_{2},\varphi\right)$ holds, then**

[TABLE]

We omit the proof of Theorem 28 as it is a natural consequence of Theorem 27.

Remark 1**.**

If we take $\frac{f_{1}}{f_{2}}$ instead of $f_{1}\cdot f_{2}$ and $\frac{g_{1}}{g_{2}}$ instead of $g_{1}\cdot g_{2}$ where $\frac{f_{1}}{f_{2}}$ is meromorphic and $\frac{g_{1}}{g_{2}}$ is entire function, and the other conditions of Theorem 25, Theorem 26, Theorem 27 and Theorem 28 remain the same, then conclusion of Theorem 25, Theorem 26, Theorem 27 and Theorem 28 remains valid.