On uniqueness of two meromorphic functions sharing a small function

TL;DR

This paper studies the uniqueness of meromorphic functions sharing a small function, extending previous results by considering differential polynomials and correcting earlier errors, with implications for future research.

Contribution

It extends and improves existing uniqueness results for meromorphic functions sharing small functions, including corrections of prior mistakes and new open questions.

Findings

Extended the class of functions sharing small functions.

Reduced the lower bound of n in the main theorems.

Corrected previous errors in the literature.

Abstract

In this paper, we have investigated the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing a small function. Our results radically extended and improved the results of Bhoosnurmath-Pujari and Harina - Anand not only by sharing small function instead of fixed point but also reducing the lower bound of $ n $. The authors Harina-Anand made plenty of mistakes in their paper. We have corrected all of them in a more convenient way. At last some open questions have been posed for further study in this direction.

Equations

$$f(z)=\frac{(n+2)}{(n+1)}\frac{e^z+\dots +e^{(n+1)z}}{1+e^z+\dots +e^{(n+1)z}}$$

$$f(z)=\frac{(n+2)}{(n+1)}\frac{1+e^z+\dots +e^{nz}}{1+e^z+\dots +e^{(n+1)z}}.$$

$$g=\frac{(n+2)(1-h^{n+1})}{(n+1)(1-h^{n+2})},f=\frac{(n+2)h(1-h^{n+1})}{(n+1)(1-h^{n+2})},$$

$$\mathcal{P}(w)=w^{n+m}+\dots +a_n{w}^n+\dots +a_0=a_{n+m}\prod _{i=1}^s(w-w_{p_i}{)}^{p_i}$$

$$T(r,f_1)<\sum _{i=1}^3{N}_2\left(r,\frac{1}{f_i}\right)+\sum _{i=1}^3\overline{N}(r,f)+o(T(r)),$$

$$T(r,f_1)⩽\overline{N}(r,f_1)+\overline{N}\left(r,\frac{1}{f_1}\right)+\overline{N}\left(r,\frac{1}{f_2}\right)+S(r,f_1).$$

$$N\left(r,\frac{1}{f^{(k)}}\right)⩽N(r,\frac{1}{f})+k\overline{N}(r,f)+S(r,f).$$

$$T(r,P(f))=nT(r,f)+S(r,f).$$

$$T(r,g_{\ast })⩽\left(\frac{p+q+2}{p-6}\right)T(r,f_{\ast })+S(r,g_{\ast })$$

$$T\left(r,{\mathcal{P}}_{\ast }(g)g_{\ast }^{\prime }\right)$$

$$(p+q)T(r,g)$$

$$\overline{N}\left(r,\frac{1}{f_{\ast }^p(f_{\ast }-1{)}^q{f}_{\ast }^{\prime }-\alpha }\right)$$

$$(p+q)T(r,g_{\ast })$$

$$T(r,g)⩽\left(\frac{p+q+2}{p-6}\right)T(r,f)+S(r,g),$$

$$T(r,g_{\ast })⩽\left(\frac{p+q+2}{p-3}\right)T(r,f_{\ast })+S(r,g_{\ast })$$

$$F(t)$$

$$[F(t){]}^{\prime }=(4b-c^2)(q-p)\sinh (q-p)t-4bq\sinh qt,$$

$$[F(t){]}^{\prime \prime }=(4b-c^2)(q-p{)}^2\cosh (q-p)t-4b{q}^2\cosh qt,$$

$$[F(t){]}^{\prime \prime \prime }=(4b-c^2)(q-p{)}^3\sinh (q-p)t-4b{q}^3\sinh qt$$

$$[F(t){]}^{(iv)}=(4b-c^2)(q-p{)}^4\cosh (q-p)t-4b{q}^4\cosh qt.$$

$$(4b-c^2)\cosh (q-p)t-4b\cosh qt+c^2=0$$

$$(4b-c^2)(q-p)\sinh (q-p)t-4qb\sinh qt=0.$$

$$\frac{(4b\cosh qt-c^2{)}^2}{(4b-c^2{)}^2}-\frac{16q^2{b}^2{\sinh }^{2}qt}{(4b-c^2{)}^2(q-p{)}^2}=1.$$

$$(q-p{)}^2(4b{\cosh }^{2}qt-c^2{)}^2-16q^2{b}^2({\cosh }^{2}qt-1)=(4b-c^2{)}^2(q-p{)}^2.$$

$$\displaystyle\bigg{\{}\cosh qt-1\bigg{\}}\bigg{\{}\cosh qt-\frac{a^{2}(q-p)^{2}}{2bp(p-2q)}+1\bigg{\}}=0.$$

$$\mathcal{H}\equiv \frac{{\mathcal{P}}_{\ast }(f)f_{\ast }^{\prime }-\alpha }{{\mathcal{P}}_{\ast }(g)g_{\ast }^{\prime }-\alpha }\equiv \frac{f_{\ast }^p(f_{\ast }-1{)}^q{f}_{\ast }^{\prime }-\alpha }{g_{\ast }^p(g_{\ast }-1{)}^q{g}_{\ast }^{\prime }-\alpha }.$$

$$T(r,\mathcal{H})$$

$$T(r,\mathcal{H})=O(T_{\ast }(r)).$$

Full text

1. Introduction, definitions and main results

The Nevanlinna theory mainly describes the asymptotic distribution of solutions of the equation $f(z)=w$, as $w$ varies. At the outset, we assume that readers are familiar with the basic Nevanlinna Theory [12]. First we explain the general sharing notion. Let $f$ and $g$ be two non-constant meromorphic functions in the complex plane $\mathbb{C}$. Two meromorphic functions $f$ and $g$ are said to share a value $w\in\mathbb{C}\cup\{\infty\}$ $IM$ (ignoring multiplicities) if $f$ and $g$ have the same $w$-points counted with ignoring multiplicities. If multiplicities of $w$-points are counted, then $f$ and $g$ are said to share $w$ $CM$ (counting multiplicities).

When $w=\infty$ the zeros of $f-w$ means the poles of $f$.

It is well known that if two moromorphic functions $f$ and $g$ share four distinct values $CM$, then one is Möbius Transformation of the other. In $1993$, corresponding to one famous question of Hayman [13], Yang-Hua [19] showed that similar conclusions hold for certain types of differential polynomials when they share only one value.

Recently by using the same argument as in [19], Fang-Hong [10] the following result was obtained.

Theorem A**.**

Let $f$ and $g$ be two transcendental entire functions, $n\geqslant 11$, an integer. If $f^{n}(f-1)f^{\prime}$ and $g^{n}(g-1)g^{\prime}$ share $1$ $CM$, then $f\equiv g$.

The following example shows that in Theorem A one simply can not replace “entire” by “meromorphic ” functions.

Example 1.1**.**

Let

[TABLE]

and

[TABLE]

It is clear that $f(z)=e^{z}g(z)$. Also $f^{n}(f-1)f^{\prime}$ and $g^{n}(g-1)g^{\prime}$ share $1$ $CM$ but note that $f\not\equiv g$.

In $2004$, Lin-Yi [14] extended Theorem A and obtained the following results.

Theorem B**.**

[14]** Let $f$ and $g$ be two transcendental entire functions, $n\geqslant 7$ an integer. If $f^{n}(f-1)f^{\prime}$ and $g^{n}(g-1)g^{\prime}$ share $z$ $CM$, then $f\equiv g$.

Theorem C**.**

[14]** Let $f$ and $g$ be two transcendental meromorphic functions, $n\geqslant 12$ an integer. If $f^{n}(f-1)f^{\prime}$ and $g^{n}(g-1)g^{\prime}$ share $z$ $CM$, then either $f\equiv g$ or

[TABLE]

where $h$ is a non-constant meromorphic function.

Theorem D**.**

[14]** Let $f$ and $g$ be two transcendental meromorphic functions, $n\geqslant 13$ an integer. If $f^{n}(f-1)^{2}f^{\prime}$ and $g^{n}(g-1)^{2}g^{\prime}$ share $z$ $CM$, then $f\equiv g$.

To improve all the above mentioned results, natural questions arise as follows.

Question 1.1**.**

Keeping all other conditions intact, is it possible to reduce further the lower bounds of $n$ in the above results ?

Question 1.2**.**

Is it also possible to replace the transcendental meromorphic (entire) functions by a more general class of meromorphic (entire) functions in all the above mentioned results ?

In $2013$, Bhoosnurmath-Pujari [9], answered the above questions affirmatively and obtained the following results.

Theorem E**.**

[9]** Let $f$ and $g$ be two non-constant meromorphic functions, $n\geqslant 11$ be an integer. If $f^{n}(f-1)f^{\prime}$ and $g^{n}(g-1)g^{\prime}$ share $z$ $CM$, $f$ and $g$ share $\infty$ $IM$, then either $f\equiv g$ or

[TABLE]

where $h$ is a non-constant meromorphic function.

Theorem F**.**

[9]** Let $f$ and $g$ be two non-constant meromorphic functions, $n\geqslant 12$ an integer. If $f^{n}(f-1)^{2}f^{\prime}$ and $g^{n}(g-1)^{2}g^{\prime}$ share $z$ $CM$, $f$ and $g$ share $\infty$ $IM$, then $f\equiv g$.

Theorem G**.**

[9]** Let $f$ and $g$ be two non-constant entire functions, $n\geqslant 7$ be an integer. If $f^{n}(f-1)f^{\prime}$ and $g^{n}(g-1)g^{\prime}$ share $z$ $CM$, then $f\equiv g$.

In this direction, for the purpose of extension Theorem E and F, one may ask the following question.

Question 1.3**.**

Keeping all other conditions intact in Theorem E, F and G, is it possible to replace respectively $f^{n}(f-1)f^{\prime}$ and $g^{n}(g-1)g^{\prime}$ by $f^{n}(f-1)^{m}f^{\prime}$ and $g^{n}(g-1)^{m}g^{\prime}$ ?

Next the following question is inevitable.

Question 1.4**.**

Is it possible to omit the second conclusions of Theorems C and E ?

In $2016$, Waghmore-Anand [16] answer the Questions 1.3 and 1.4  affirmatively and obtained the following results.

Theorem H**.**

[16]** Let $f$ and $g$ be two non-constant meromorphic functions, $n\geqslant m+10$ be an integer. If $f^{n}(f-1)^{m}f^{\prime}$ and $g^{n}(g-1)^{m}g^{\prime}$ share $z$ $CM$, $f$ and $g$ share $\infty$ $IM$, then $f\equiv g$.

Theorem I**.**

[16]** Let $f$ and $g$ be any two non-constant entire functions, $n\geqslant m+6$ an integer. If $f^{n}(f-1)^{m}f^{\prime}$ and $g^{n}(g-1)^{m}g^{\prime}$ share $z$ $CM$, then $f\equiv g$.

Note 1.1**.**

We see that in the results of Waghmore - Anand, for $m=2$, Theorem H reduces to *Theorem F * and for $m=1$, Theorem I reduces to *Theorem G *.

Remark 1.1*.*

We notice that in the proof of Theorem H and hence in the case of Theorem I also, there are plenty of mistakes made by the authors Waghmore-Anand [16]. We mention below few of them.

1. (i)

In [16, page-947], just before Case 2, the authors obtained that the coefficient of $T(r,g)$ is $(n-m-8)$, while actually it will be $(n+m-8)$. 2. (ii)

In [16, page-948], just before Case 3, the authors finally obtained that “$h^{n+m}-1=0$, $h^{n+1}-1=0$, which imply $h=1$”. Note that this possible only when $gcd(n+m,n+1)=1$ but which is not true if one consider some suitable value of $n$ and $m$. For example if we choose $n=3$ and $m=5$, we note that $gcd(n+m,n+1)=gcd(8,4)=4\neq 1$. 3. (iii)

We observe that in [16, equation (49), page-950], the coefficient of $T(r,g)$ is $\displaystyle\frac{m}{n+m-1}$ while actually it should be $\displaystyle\frac{m}{n+m+1}$.

Thus we see that study on derivative or differential polynomial has a long history, several authors have been main engaged to find a possible relationship or certain forms of a function $f$ when it shares small function with its derivative or differential polynomial (see [1] - [8].)

In this paper, our aim is to correct all the mistakes made by Waghmore-Anand [16] and at the same time to get an improved and extended version results of all the above mentioned Theorems A - I.

To this end, throughout the paper, we will use the following transformations. Let

[TABLE]

where $a_{j}(j=0,1,2,\ldots,n+m-1)$ and $w_{p_{i}}(i=1,2,...,s)$ are distinct finite complex numbers and $2\leqslant s\leqslant n+m$ and $p_{1},p_{2},\ldots,p_{s}$, $s\geqslant 2$, $n$, $m$ and $k$ are all positive integers with $\displaystyle\sum_{i=1}^{s}p_{i}=n+m$. Also let $p>\displaystyle\max_{p\neq p_{i},i=1,\ldots,r}\{p_{i}\}$, $r=s-1$, where $s$ and $r$ are two positive integers.

Let $\mathcal{Q}(w_{*})=\displaystyle\prod_{i=1}^{s-1}(w_{*}+w_{p}-w_{p_{i}})^{p_{i}}=b_{q}w_{*}^{q}+b_{q-1}w_{*}^{q-1}+\ldots+b_{0}$, where $w_{*}=w-w_{p}$, $q=n+m-p$. So it is clear that $\mathcal{P}(w)=w_{*}^{p}\mathcal{Q}(w_{*})$

In particular, if we choose $b_{i}=\displaystyle(-1)^{i}\;{{}^{q}C_{i}}$, for $i=0,1,\ldots,q$. Then we get, easily $\mathcal{P}_{*}(w)=w_{*}^{p}(w_{*}-1)^{q}$.

Note that if $w_{p}=0$ and $p=n$, then we get $w=w_{*}$ and $\mathcal{P}_{*}(w)=w^{n}(w-1)^{m}$.

Observing all the above mentioned results, we note that $h^{n}(h-1)h^{\prime}$ or $h^{n}(h-1)^{2}h^{\prime}$ $(h=f\;\text{or}\;g)$ are a special form of $h^{n}(h-1)^{m}h^{\prime}$, $m\geqslant 1$ be an integer.

So for the improvements and extensions of the above mentioned results further to a large extent, the following questions are inevitable.

Question 1.5**.**

Is it possible to replace $f^{n}(f-1)^{m}f^{\prime}$ and $g^{n}(g-1)^{m}g^{\prime}$ by a more general expressions of the form $\mathcal{P}_{*}(f)f_{*}^{\prime}=f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}=g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ respectively in all the above mentioned results ?

If the answer of the Question 1.5  is found to be affirmative, then one my ask the following questions.

Question 1.6**.**

Is it possible to reduce further the lower bounds of $n$ in Theorems $E$, $F$, $G$ and $H$ ?

Question 1.7**.**

Is it also possible to replace sharing $z$ $CM$ by sharing $\alpha(z)$ $CM$ in Theorem G and H ?

Answering all the above mentioned questions affirmatively is the main motivation of writing this paper.

Following two theorems are the main results of this paper improving and extending all the above mentioned results to a more convenient way and compact form.

Theorem 1.1**.**

Let $f$ and $g$ hence $f_{*}=f-w_{p}$ and $g_{*}=g-w_{p}$, $w_{p}\in\mathbb{C}$ be any two non-constant non- entire meromorphic functions, $n\geqslant q+9$, $q$ $\in\mathbb{N}$, be an integer. If $\mathcal{P}_{*}(f)f_{*}^{\prime}=f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}=g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ share $\alpha\equiv\alpha(z)$ $(\not\equiv 0,\infty)$ $CM$, $f_{*}$ and $g_{*}$ share $\infty$ $IM$, then $f\equiv g$.

Theorem 1.2**.**

Let $f$ and $g$ hence $f_{*}=f-w_{p}$ and $g_{*}=g-w_{p}$, $w_{p}\in\mathbb{C}$ be any two non-constant entire functions, $n\geqslant q+5$, $q$ $\in\mathbb{N}$, be an integer. If $\mathcal{P}_{*}(f)f_{*}^{\prime}=f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}=g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ share $\alpha\equiv\alpha(z)$ $(\not\equiv 0,\infty)$ $CM$, then $f\equiv g$.

2. Some lemmas

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

Lemma 2.1**.**

[17]** Let $f_{1}$, $f_{2}$ and $f_{3}$ be non constant meromorphic functions such that $f_{1}+f_{2}+f_{3}=1$. If $f_{1}$, $f_{2}$ and $f_{3}$ are linearly independent, then

[TABLE]

where T(r)=\displaystyle\max_{1\leqslant i\leqslant 3}\bigg{\{}T(r,f_{i})\bigg{\}} and $r\not\in E$.

Lemma 2.2**.**

[20]** Let $f_{1}$ and $f_{2}$ be two non-constant meromorphic functions. If $c_{1}f_{1}+c_{2}f_{2}=c_{3}$, where $c_{i}$, $i=1,2,3$ are non-zero constants, then

[TABLE]

Lemma 2.3**.**

[20]** Let $f$ be a non-constant meromorphic function and $k$ be a non-negative integer, then

[TABLE]

Lemma 2.4**.**

[22]** Suppose that $f$ is a non-constant meromorphic function and $P(f)=a_{n}f^{n}+a_{n-1}f^{n-1}+\ldots+a_{1}f+a_{0}$, where $a_{n}(\not\equiv 0)$, $a_{n-1},\ldots,a_{1},a_{0}$ are small meromorphic functions of $f(z)$. Then

[TABLE]

Lemma 2.5**.**

[18]** Let $f_{1}$, $f_{2}$ and $f_{3}$ be three meromorphic functions satisfying $\displaystyle\sum_{i=1}^{3}f_{i}=1,$ then the functions $g_{1}=-\displaystyle\frac{f_{1}}{f_{2}}$, $g_{2}=\displaystyle\frac{1}{f_{2}}$ and $g_{3}=-\displaystyle\frac{f_{1}}{f_{2}}$ are linearly independent when $f_{1}$, $f_{2}$ and $f_{3}$ are linearly independent.

Lemma 2.6**.**

Let $f$ and $g$ and hence $f_{*}=f-w_{p}$ and $g_{*}=g-w_{p}$ be two non-constant meromorphic functions and $\alpha\equiv\alpha(z)$ $(\not\equiv 0,\infty)$ be a small function of $f$ and $g$. If $\mathcal{P}_{*}(f)f_{*}^{\prime}=f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}=g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ share $\alpha$ $CM$ and $p\geqslant 7$, then

[TABLE]

Proof.

Applying Second Fundamental Theorem on $\mathcal{P}_{*}(g)g_{*}^{\prime}$, we get

[TABLE]

Next by applying First fundamental Theorem,

[TABLE]

After combining (2) and (2), we get

[TABLE]

Again since $S(r,g_{*})=T(r,\alpha)=S(r,f_{*})$, so we must have

[TABLE]

By using (2.6) in (2.5), we get

[TABLE]

i.e.,

[TABLE]

where $p\geqslant 7.$ ∎

Lemma 2.7**.**

Let $f$ and $g$ and hence $f_{*}=f-w_{p}$ and $g_{*}=g-w_{p}$ be two non-constant entire functions and $\alpha\equiv\alpha(z)$ $(\not\equiv 0,\infty)$ be a small function of $f$ and $g$. If $\mathcal{P}_{*}(f)f_{*}^{\prime}=f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}=g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ share $\alpha$ $CM$ and $p\geqslant 5$, then

[TABLE]

Proof.

Since $f$ and $g$ both are entire functions, so we must have $\overline{N}(r,f)=0=\overline{N}(r,g)$.

Proceeding exactly as in the line of the proof of Lemma 2.6, we can prove the lemma. ∎

Lemma 2.8**.**

Let $\Psi(z)=c^{2}(z^{p-q}-1)^{2}-4b(z^{p-2q}-1)(z^{p}-1)$ , where $b,c\in\mathbb{C}-\{0\},\displaystyle\frac{c^{2}}{4b}=\displaystyle\frac{p(p-2q)}{(p-q)^{2}}\neq 1,\;$, then $\Psi(z)$ has exactly one multiple zero of multiplicity $4$ which is $1$.

Proof.

We claim that $\Psi(1)=0$ with multiplicity $4$ and all other zeros of $\Psi(w)$ are simple. Let $F(t)=\displaystyle\frac{1}{2}\Psi(e^{t})e^{(q-p)t}.$ Then

[TABLE]

Next we see that for $t=0$, $F(t)=0$, $[F(t)]^{\prime}=0$, $[F(t)]^{\prime\prime}=0$ since $\displaystyle\frac{c^{2}}{4b}=\frac{p(p-2q)}{(p-q)^{2}}$ and $[F(t)]^{\prime\prime\prime}=0$ but $[F(t)]^{(iv)}\neq 0$ where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Therefore it is clear that $F(0)=0$ with multiplicity $4$ and hence $\Psi(1)=0$ with multiplicity $4$.

Next we suppose that $\Psi(w)=0=\Psi^{\prime}(w)$, for some $w\in\mathbb{C}$. Then $F(t)=0=F^{\prime}(t)$ for every $t$ satisfying $e^{qt}=w$. Now from $F(t)=0$ and $F^{\prime}(t)=0$, we obtained respectively

[TABLE]

and

[TABLE]

Since $\cosh^{2}(q-p)t-\sinh^{2}(q-p)t=1$, so from (2.5) and (2.6), we get

[TABLE]

i.e.,

[TABLE]

i.e.,

[TABLE]

Since $\displaystyle\frac{c^{2}}{4b}=\displaystyle\frac{p(p-2q)}{(p-q)^{2}}$, then $\displaystyle\frac{c^{2}(q-p)^{2}}{2bq(q-2p)}=2$, so we see that the equation (2.7) reduces to \bigg{\{}\cosh qt-1\bigg{\}}^{2}=0. i.e., we get $e^{qt}=1=w.$ ∎

3. Proofs of the theorems

Proof of Theorem 1.1.

Since $\mathcal{P}_{*}(f)f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}$ share $\alpha\equiv\alpha(z)$ $CM$, $f$ and $g$ share $\infty$ $IM$, so we suppose that

[TABLE]

Then from (2.6) and (3.1), we get

[TABLE]

where $T_{*}(r)=\max\{T(r,f_{*}),T(r,g_{*})\}$ and $S_{*}(r)=\max\{S(r,f_{*}),S(r,g_{*})\}$.

i.e.,

[TABLE]

Again from (3.1), we see that the zeros and poles of $\mathcal{H}$ are multiple and hence

[TABLE]

Let $f_{1}=\displaystyle\frac{f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}}{\alpha}$,     $f_{2}=\mathcal{H}$     and     $f_{3}=-\mathcal{H}\displaystyle\frac{g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}}{\alpha}$.

Thus we get $f_{1}+f_{2}+f_{3}=1$. Next we denote $T(r)=\max\{T(r,f_{1}),T(r,f_{2}),T(r,f_{3})\}.$

We have,

[TABLE]

[TABLE]

So we have $T(r,f_{i})=O(T_{*}(r))$ for $i=1,2,3$ and hence $S(r,f_{*})+S(r,g_{*})=o(T_{*}(r))$.

Next we discuss the following cases.

Case 1. Suppose none of $f_{2}$ and $f_{3}$ is a constant. If $f_{1}$, $f_{2}$ and $f_{3}$ are linearly independent, then by Lemma 2.1 and 2.4, we have

[TABLE]

We see that $N_{2}\left(r,\displaystyle\frac{1}{\mathcal{H}}\right)\leqslant 2\overline{N}\left(r,\displaystyle\frac{1}{\mathcal{H}}\right)\leqslant 2\overline{N}_{L}(r,g_{*})$,    $\overline{N}(r,\mathcal{H})\leqslant\overline{N}_{L}(r,f)$.

Again since $\overline{N}_{L}(r,f_{*})=0=\overline{N}_{L}(r,g_{*})$ and note that $\overline{N}(r,f_{*})=\overline{N}(r,g_{*})$, so using all this facts, we get from (3) that

[TABLE]

Let $z_{0}$ be a zero of $f_{*}$ of multiplicity $r$, then $z_{0}$ is a zero of $f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ of multiplicity $pr+r-1\geqslant 3$. Thus we have

[TABLE]

Similarly, we get

[TABLE]

Let

[TABLE]

and

[TABLE]

By Lemma 2.4, we have

[TABLE]

It is clear $\mathcal{F}^{\prime}=\alpha f_{1}$. So we have

[TABLE]

By using First fundamental Theorem and (3.8), we obtained

[TABLE]

where $a_{i}$ $(i=1,2,\ldots,q)$ are the roots of the algebraic equation

[TABLE]

Using (3) - (3.8) in (3), we get

[TABLE]

i.e.,

[TABLE]

i.e.,

[TABLE]

Let $g_{1}=-\displaystyle\frac{f_{3}}{f_{2}}=\frac{g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}}{\alpha}$,   $g_{2}=\displaystyle\frac{1}{f_{2}}=\frac{1}{\mathcal{H}}$   and   $g_{3}=-\displaystyle\frac{f_{1}}{f_{2}}=-\frac{f_{*}^{p}(f_{*}-1)^{q}f^{\prime}}{\alpha\mathcal{H}}.$

Then we get $g_{1}+g_{2}+g_{3}=1$. By Lemma 2.5, $g_{1}$, $g_{2}$ and $g_{3}$ are linearly independent since $f_{1}$, $f_{2}$ and $f_{3}$ are linearly independent. Proceeding exactly same way as done in above, we get

[TABLE]

Let $T_{*}(r)=\max\{T(r,f_{*}),T(r,g_{*})\}$. After combining (3.10) and (3.11), we get

[TABLE]

which contradicts $p\geqslant q+9.$

Thus $f_{1}$, $f_{2}$ and $f_{3}$ must be linearly dependent. Therefore there exists three constants $c_{1}$, $c_{2}$ and $c_{3}$, at least one of them are non-zero such that

[TABLE]

Subcase 1.1. If $c_{1}=0$, $c_{2}\neq 0$ and $c_{3}\neq 0$, then from (3.12) we get $\displaystyle f_{3}=-\frac{c_{2}}{c_{3}}f_{2}$ which implies $g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}=\displaystyle\frac{c_{2}}{c_{3}}\alpha.$

On integrating, we get

[TABLE]

where $c$ is an arbitrary constant.

Thus we see that

[TABLE]

i.e.,

[TABLE]

Since $p\geqslant q+9$, so we get a contradiction.

Subcase 1.2. Let $c_{1}\neq 0$. Then from (3.12), we get

[TABLE]

After substituting this in the relation $f_{1}+f_{2}+f_{3}=1$, we get

[TABLE]

where $(c_{1}-c_{2})(c_{1}-c_{3})\neq 0$. So we get

[TABLE]

Again we see that

[TABLE]

Next applying Lemma 2.2 to the equation (3.14), we get

[TABLE]

So combining the above two we get,

[TABLE]

By applying Lemmas 2.3, 2.4 and (3.15), we have

[TABLE]

which contradicts $p\geqslant q+9$.

Subcase 2. If $f_{2}=k$, where $k$ is a constant.

Subcase 2.1 If $k\neq 1$, then from the relation $f_{1}+f_{2}+f_{3}=1$, we get

[TABLE]

Next we apply Lemma 2.2 to the equation (3.16), we get

[TABLE]

By applying Lemma 2.3, 2.4 and using equation (3), we get

[TABLE]

i.e.,

[TABLE]

Using Lemma 2.6, we get

[TABLE]

which contradicts $p\geqslant q+9$.

Subcase 2.2 Let $k=1$ i.e., $\mathcal{H}=1$ i.e.,

[TABLE]

On integrating, we get

[TABLE]

where $c$ is an arbitrary constant. i.e.,

[TABLE]

Subcase 2.2.1 Let if possible $c\neq 0$. Next we get

[TABLE]

We have,

[TABLE]

Similarly, we get $\overline{\displaystyle}N\left(r,\displaystyle\frac{1}{\mathcal{G}}\right)\leqslant(q+1)\;T(r,g_{*}).$

Again note that $\overline{N}(r,\mathcal{F})=\overline{N}(r,f_{*})\leqslant T(r,f_{*}).$ Again

[TABLE]

[TABLE]

Thus

[TABLE]

Similarly

[TABLE]

Therefore

[TABLE]

since $p\geqslant q+9$, which is a contradiction.

Subcase 2.2.2 Thus we get $c=0$. Thus we get

[TABLE]

Let $h=\displaystyle\frac{f_{*}}{g_{*}}$. Then substituting in (3.19), we get

[TABLE]

Subcase 2.2.2.1. If $h$ is a non-constant, then using Lemma 2.8 and proceeding exactly same way as done in [15, p-1272], we arrive at a contradiction.

Subcase 2.2.2.2. Let $h$ is constant, then from (3.20), we get $h^{p+q+1}-1=0$, $h^{p+q}-1=0$, $\ldots$, $h^{p+1}-1=0$. i.e., $h^{d}-1=0$, where $d=gcd(p+q+1,p+q,\ldots,p+1)=1.$ i.e., $h=1$.

Hence $f_{*}\equiv g_{*}.$ i.e., $f\equiv g$.

Subcase 3. Suppose $f_{3}=c$, where $c$ is a constant.

Subcase 3.1. If $c\neq 1$, then from the relation $f_{1}+f_{2}+f_{3}=1$, we get

[TABLE]

Applying Lemma 2.2 to the above equation, we get

[TABLE]

Using Lemma 2.3, 2.4 and (3), we have

[TABLE]

Next by applying Lemma 2.6, we get

[TABLE]

which contradicts $p\geqslant q+9$.

Subcase 3.2. Let $c=1$. Then from (3.21), we get

[TABLE]

Let $z_{0}$ be a zero of $f_{*}$ of order $r_{0}$. Then from (3.23), we see that $z_{0}$ is a pole of $g_{*}$ of order $s_{0}$ (say). Then from (3.23), we get $pr_{0}+r_{0}-1=ps_{0}+qs_{0}+s_{0}+1.$ i.e., $(p+1)(r_{0}-s_{0})=qs_{0}+2\geqslant p+1.$ i.e.,

[TABLE]

Again let $z_{1}$ be a zero of $f_{*}-1$ of order $r_{1}$. Then from (3.23), we see that $z_{1}$ will be a pole of $g_{*}$ of order $s_{1}$ (say). So we have $r_{1}+r_{1}-1=ps_{1}+qs_{1}+s_{1}+1.$ i.e.,

[TABLE]

Let $z_{2}$ be a zero of $f_{*}^{\prime}$ of order $r_{2}$ which are not the zero of $f_{*}(f_{*}-1)$, so from (3.23) we see that $z_{2}$ will be a pole of $g_{*}$ of order $s_{2}$ (say). Then from (3.23), we get $r_{2}=ps_{2}+qs_{2}+s_{2}+1$. i.e.,

[TABLE]

The similar explanations hold for the zeros of $g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ also. Next we see from (3.23), we have

[TABLE]

i.e.,

[TABLE]

By applying Second Fundamental Theorem, we get

[TABLE]

Similarly, we get

[TABLE]

From (3) and (3), we get

[TABLE]

i.e.,

[TABLE]

which contradicts $p\geqslant q+9$. ∎

Proof of Theorem 1.2.

Since $f_{*}$ and $g_{*}$ both are non-constant entire functions, then we may consider the followings two cases.

Case 1. Let $f_{*}$ and $g_{*}$ are two transcendental entire functions. Then it is clear that $\overline{N}(r,f_{*})=S(r,f_{*})$ and $\overline{N}(r,g_{*})=S(r,g_{*})$. With this the rest of the proof can be carried out in the line of the proof of Theorem 1.1.

Case 2. Let $f_{*}$ and $g_{*}$ both are polynomials. Since $f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ share $\alpha$ $CM$, then we must have

[TABLE]

where $\kappa$ is a non-zero constant.

Subcase 2.1. Suppose $\kappa\neq 1$, then from (3.26), we get

[TABLE]

Applying Lemma 2.2, we get

[TABLE]

Using Lemmas 2.3, 2.4 and (3.27), we get

[TABLE]

i.e.,

[TABLE]

Using Lemma 2.7, we get

[TABLE]

which contradicts $p\geqslant q+5$.

Subcase 2.2. Let $\kappa=1$. So from (3.27), we get

[TABLE]

Next proceeding exactly same way as done in Subcase 1.3.2  in the proof of Theorem 1.1, we get $f\equiv g$. ∎

4. Concluding remarks and some open questions

If we replace the condition $``f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ share $\alpha(z)$ $CM$” by the condition $``f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ share $z$ $CM$ ”, then the conclusions of Theorems 1.1 and 1.2 still hold.

Thus we get the following results

Theorem 4.1**.**

Let $f$ and $g$ hence $f_{*}=f-w_{p}$ and $g_{*}=g-w_{p}$, $w_{p}\in\mathbb{C}$ be any two non-constant non- entire meromorphic functions, $n\geqslant q+9$, $q$ $\in\mathbb{N}$, be an integer. If $\mathcal{P}_{*}(f)f_{*}^{\prime}=f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}=g_{*}^{p}(g_{*}-1)^{q}f_{*}^{\prime}$ share $z$ $CM$, then $f\equiv g$.

Theorem 4.2**.**

Let $f$ and $g$ hence $f_{*}=f-w_{p}$ and $g_{*}=g-w_{p}$, $w_{p}\in\mathbb{C}$ be any two non-constant entire functions, $n\geqslant q+5$, $q$ $\in\mathbb{N}$, be an integer. If $\mathcal{P}_{*}(f)f_{*}^{\prime}=f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $\mathcal{P}_{*}(g)g_{*}^{\prime}=g_{*}^{p}(g_{*}-1)^{q}f_{*}^{\prime}$ share $z$ $CM$, then $f\equiv g$.

Note 4.1**.**

If we choose $q=m$, $w_{p}=0$, then since $p=n+m-q$ and $f_{*}=f-w_{p}$, so we get $p=n$ and $f_{*}=f$, respectively. With this we see that $n\geqslant m+9$ in Theorem 4.1 and $n\geqslant m+5$ in Theorem 4.2.

So from the above note, we observe that Theorem 4.1 and Theorem 4.2 are the direct improvement as well as extension of Theorem H and I respectively.

Remark 4.1*.*

We see from Note 4.1 that for $m=1$ and $m=2$, we get $n\geqslant 10$ and $n\geqslant 11$ respectively in Theorem 4.1 which is a direct improvement of Theorem E and F.

Remark 4.2*.*

For $m=1$, we see from Note 4.1 that $n\geqslant 6$ in Theorem 4.2 which is a direct improvement of Theorem G.

Next for further research in this direction, one my glance over the following remarks.

Remark 4.3*.*

What worth noticing fact is that in [16, equation (39)], there is no term which is absent in the expression. So, for the case of $h$ is constant, [16, equation (40)] implies $h^{d}-1=0$, where $d=gcd(n+m+1,n+m,\ldots,n+1)=1.$ i.e., $h=1$ and hence $f\equiv g$. But if we replace $(f-1)^{m}$ in the expression $f^{n}(f-1)^{m}f^{\prime}$ by a more general expression $f^{n}P_{m}(f)f^{\prime}$, where $P_{m}(f)=a_{m}f^{m}+a_{m-1}f^{m-1}+\ldots+a_{1}f+a_{0}$, $a_{i}\in\mathbb{C}$, for $i=0,1,\ldots,m$. It is not always possible to handle the case of $h$ is constant. If somehow one can do that, then from the case of $h$ is constant, $h^{d}-1=0$, where $d=gcd(n+m+1,n+m,\ldots,n+1)\neq 1$ in general. So we can’t obtained $f\equiv g$ in general.

Based on the above observations, we next pose the following open questions.

Question 4.1**.**

Is it possible to reduce further the lower bounds of $p$ in Theorem 1.1 and Theorem 1.2 ?

Question 4.2**.**

To get the uniqueness between $f$ and $g$ is it possible to replace $f_{*}^{p}(f_{*}-1)^{q}f_{*}^{\prime}$ and $g_{*}^{p}(g_{*}-1)^{q}g_{*}^{\prime}$ respectively by $f_{*}^{p}P_{m}(f_{*})f_{*}^{\prime}$ and $g_{*}^{p}P_{m}(g_{*})g_{*}^{\prime}$, where $P_{m}(f_{*})=P_{m}(f)=a_{m}f_{*}^{m}+a_{m-1}f_{*}^{m-1}+\ldots+a_{1}f_{*}+a_{0}$ in Theorem 1.1 and Theorem 1.2 ?