On the value distribution of a Differential Monomial and some normality criteria

TL;DR

This paper investigates the value distribution of differential monomials of transcendental meromorphic functions and establishes a normality criterion for families of analytic functions based on differential inequalities.

Contribution

It provides a quantitative estimate of the characteristic function related to differential monomials and proves a new normality criterion involving differential inequalities.

Findings

Quantitative estimation of the characteristic function T(r, f)

Normality criterion for families of analytic functions

Conditions involving differential monomials and zeros

Abstract

Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $\varphi(\not\equiv 0,\infty)$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r, f)$ in terms of $N\left(r,\frac{1}{M[f]-\varphi(z)}\right)$ as well as $\ol{N}\left(r,\frac{1}{M[f]-\varphi(z)}\right)$, where $M[f]$ is the differential monomial, generated by $f$.\par Moreover, we prove one normality criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq1)$, $q_{0}(\geq 3)$, $q_{i}(\geq0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq1)$ be positive integers. If for each $f\in \mathscr{F}$, $f$ has only zeros of multiplicity at least $k$, and $f^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}\not=1$, then $\mathscr{F}$ is normal on domain $D$.

Equations

$$S(r,f)=o(T(r,f))\text{as}r\to \infty ,r\notin E.$$

$$T(r,f)\leq 6N\bigg{(}r,\frac{1}{f^{2}f^{\prime}-1}\bigg{)}+S(r,f).$$

$$T(r,f)\leq 6\overline{N}\bigg{(}r,\frac{1}{\phi f^{2}f^{{}^{\prime}}-1}\bigg{)}+S(r,f).$$

$$T(r,f)\leq 6N\bigg{(}r,\frac{1}{f^{2}f^{(k)}-1}\bigg{)}+S(r,f).$$

$$(p+n)T(r,f)\le \overline{N}(r,\infty ;f)+\overline{N}(r,0;f)+p{N}_k^{\ast }(r,0;f)+\overline{N}(r,a;\psi )+S(r,f).$$

$$T(r,f)\leq\frac{1}{l-2}\overline{N}\bigg{(}r,\frac{1}{f^{l}(f^{(k)})^{n}-a}\bigg{)}+S(r,f).$$

$$T(r,f)\leq 2\overline{N}\bigg{(}r,\frac{1}{f^{2}f^{(k)}-1}\bigg{)}+S(r,f).$$

$$T(r,f)\leq M\overline{N}\bigg{(}r,\frac{1}{f^{2}f^{(k)}-1}\bigg{)}+S(r,f),$$

$$M[f]=a(z)(f{)}^{q_0}(f^{\prime }{)}^{q_1}...(f^{(k)}{)}^{q_k}$$

$$\mu T(r,f)$$

$$T(r,f)$$

$$(q-2-\epsilon )T(r,f)\le \sum _{i=1}^q\overline{N}(r,a_i,f)+S(r,f),$$

$$N\left(r,\frac{g^{\prime }}{g}\right)-N\left(r,\frac{g}{g^{\prime }}\right)=\overline{N}(r,g)+N\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{g^{\prime }}\right).$$

$$\underset{r\to \infty }{\lim }\frac{T(r,f)}{\log r}=\infty .$$

$${\left(\frac{1}{f}\right)}^{\mu }=a(z){\left(\frac{f^{\prime }}{f}\right)}^{q_1}{\left(\frac{f^{\prime \prime }}{f}\right)}^{q_2}\dots {\left(\frac{f^{(k)}}{f}\right)}^{q_k}\frac{1}{M[f]}.$$

$$\mu T(r,f)$$

$$T\bigg{(}r,M[f]\bigg{)}=O(T(r,f))~{}~{}\text{and}~{}~{}S\bigg{(}r,M[f]\bigg{)}=S(r,f).$$

$$\frac{1}{f^{\mu }}=\frac{M[f]}{f^{\mu }}\frac{1}{M[f]}.$$

$$\mu T(r,f)$$

$$T(r,M[f])$$

$$\le \overline{N}(r,0;M[f])+\overline{N}(r,\infty ;M[f])+\overline{N}(r,\phi (z);M[f])+\epsilon T(r,f)+S(r,f),$$

$$\mu q+1-(q\mu -{\mu }_{\ast })\le (\mu -q_0)q+(1+{\mu }_{\ast }).$$

$$\mu q+1-\{q{q}_0+(q-1)q_1+(q-2)q_2+\dots +2q_{q-2}+q_{q-1}\}$$

$$N(r,0;\phi (z))\le T(r,\phi (z))+O(1)=S(r,f),$$

$$N(r,\infty ;\phi (z))\le T(r,\phi (z))+O(1)=S(r,f),$$

$$N(r,0;f^{\mu })+\overline{N}(r,0;M[f])-N(r,0;M[f])$$

$$\mu T(r,f)$$

$$\frac{1}{f^{\mu }}=\frac{bM[f]}{f^{\mu }}-\frac{(bM[f]{)}^{\prime }}{f^{\mu }}\frac{(bM[f]-1)}{(bM[f]{)}^{\prime }}.$$

$$\mu m\left(r,\frac{1}{f}\right)$$

Full text

Moreover, we prove one normality criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq 1)$, $q_{0}(\geq 3)$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ be positive integers. If for each $f\in\mathscr{F}$, $f$ has only zeros of multiplicity at least $k$, and $f^{q_{0}}(f^{\prime})^{q_{1}}...(f^{(k)})^{q_{k}}\not=1$, then $\mathscr{F}$ is normal on domain $D$.

††footnotetext: 2010 Mathematics Subject Classification: 30D30, 30D20, 30D35.††footnotetext: Key words and phrases: Value distribution theory, Transcendental Meromorphic function, Differential Monomials, Normal family.

1. Introduction

In this paper, we use the standard notations of Nevanlinna theory ([3]). Throughout this paper, we always assume that $f$ is a transcendental meromorphic function defined in the complex plane $\mathbb{C}$. It will be convenient to let that $E$ denote any set of positive real numbers of finite linear (Lebesgue) measure, not necessarily same at each occurrence. For any non-constant meromorphic function $f$, we denote by $S(r,f)$ any quantity satisfying

[TABLE]

Definition 1.1**.**

Let $f$ be a non-constant meromorphic function. A meromorphic function $b(z)(\not\equiv 0,\infty)$ is called a “small function” with respect to $f$ if $T(r,b(z))=S(r,f)$.

Definition 1.2**.**

([18]) Let $a\in\mathbb{C}\cup\{\infty\}$. For a positive integer $k$ and for a complex constant $a$, We denote

1. i)

by $N_{k)}\left(r,a;f\right)$ the counting function of $a$-points of $f$ with multiplicity $\leq k$, 2. ii)

by $N_{(k}\left(r,a;f\right)$ the counting function of $a$-points of $f$ with multiplicity $\geq k$, 3. iii)

by $N_{k}\left(r,a;f\right)$ the counting function of the $a$-points of $f$ with multiplicity $k$.

Similarly, the reduced counting functions $\overline{N}_{k)}(r,a;f)$ and $\overline{N}_{(k}(r,a;f)$ are defined.

In 1959, Hayman proved the following theorem:

Theorem A**.**

([4]) If $f$ is a transcendental meromorphic function and $n\geq 3$, then $f^{n}f^{\prime}$ assumes all finite values except possibly zero infinitely often.

Moreover, Hayman ([4]) conjectured that the Theorem A remains valid for the cases $n=1,~{}2$. In 1979, Mues ([10]) confirmed the Hayman’s Conjecture for $n=2$ and Chen and Fang ([1]) ensured the conjecture for $n=1$ in 1995.

In 1992, Q. Zhang ([19]) gave the quantitative version of Mues’s result as follows:

Theorem B**.**

For a transcendental meromorphic function $f$, the following inequality holds :

[TABLE]

In ([15]), Theorem B was improved by Xu and Yi as

Theorem C**.**

([15]) Let $f$ be a transcendental meromorphic function and $\phi(z)(\not\equiv 0)$ be asmall function, then

[TABLE]

Also, Huang and Gu ([5]) extended Theorem B by replacing $f^{\prime}$ by $f^{(k)}$, $k(\geq 1)$ is an integer.

Theorem D**.**

([5]) Let $f$ be a transcendental meromorphic function and $k$ be a positive integer. Then

[TABLE]

Definition 1.3**.**

([8]) For a positive integer $k$, we denote $N^{\ast}_{k}(r,0;f)$ the counting function of zeros of $f$, where a zero of $f$ with multiplicity $q$ is counted $q$ times if $q\leq k$, and is counted $k$ times if $q>k$.

In 2003, I. Lahiri and S. Dewan ([8]) considered the value distribution of a differential polynomial in more general settings. They proved the following theorem.

Theorem E**.**

Let $f$ be a transcendental meromorphic function and $\alpha=\alpha(z)(\not\equiv 0,\infty)$ be a small function of $f$. If $\psi=\alpha(f)^{n}(f^{(k)})^{p}$, where $n(\geq 0)$ $p(\geq 1)$, $k(\geq 1)$ are integers, then for any small function $a=a(z)(\not\equiv 0,\infty)$ of $\psi$,

[TABLE]

The next theorem is an immediate consequence of the above theorem.

Theorem F**.**

Let $f$ be a transcendental meromorphic function and $a$ be a non zero complex constant. Let $l\geq 3$, $n\geq 1$, $k\geq 1$ be positive integers. Then

[TABLE]

In this direction, in 2009, Xu, Yi and Zhang ([13]) proved the following theorem:

Theorem G**.**

Let $f$ be a transcendental meromorphic function, and $k(\geq 1)$ be a positive integer. If $N_{1}(r,0;f)=S(r,f)$, then

[TABLE]

Later, in 2011, Xu, Yi and Zhang ([14]) removed the condition $N_{1}(r,0;f)=S(r,f)$ in above Theorem. They proved

Theorem H**.**

Let $f$ be a transcendental meromorphic function, and $k(\geq 1)$ be a positive integer.Then

[TABLE]

where $M$ is $6$ if $k=1$, or $k\geq 3$ and $M=10$ if $k=2$.

Recently, Karmakar and Sahoo([7]) considered the value distribution of the differential polynomial $f^{n}f^{(k)}-1$ where $n(\geq 2)$ and $k(\geq 1)$ are integers. Also, Xu and Ye([16]) studied the value distribution of the differential polynomial $\varphi f^{2}(f^{\prime})^{2}-1$, where $f$ is a transcendental meromorphic function, and $\varphi(z)$ is a small function of $f(z)$.

Before going to furthermore, we need to introduce the following definition:

Definition 1.4**.**

Let $f$ be nonconstant meromorphic function defined in the complex plane $\mathbb{C}$. Also, let $q_{0},q_{1},...,q_{k}$ be $(k+1)$ $(k\geq 1)$ non-negative integers and $a(z)$ be a small function of $f$. Then the expression defined by

[TABLE]

is known as differential monomial generated by $f$.

In this context, the terms $\mu:=q_{0}+q_{1}+...+q_{k}$ and $\mu_{*}:=q_{1}+2q_{2}+...+kq_{k}$ are known as the degree and weight of the differential monomial respectively.

Since the differential monomial $M[f]$ is the general form of $(f)^{q_{0}}(f^{(k)})^{q_{k}}$, so from the above discussion, the following questions are natural:

Question 1.1**.**

Does there exist positive constants $B_{1}(>0)$ and $B_{2}(>0)$ such that

1. i)

T(r,f)\leq B_{1}~{}N\bigg{(}r,\frac{1}{M[f]-1}\bigg{)}+S(r,f),   and 2. ii)

T(r,f)\leq B_{2}~{}\overline{N}\bigg{(}r,\frac{1}{M[f]-1}\bigg{)}+S(r,f)   hold?

In this paper, we deal with these questions.

2. Main Results

Theorem 2.1**.**

Let $f$ be a transcendental meromorphic function and $\varphi(z)(\not\equiv 0,\infty)$ be a small function of $f$. If $q_{0}(\geq 1)$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ are integers. Then

[TABLE]

Remark 2.1**.**

Clearly Theorem 2.1 extends Theorem E.

Corollary 2.1**.**

Let $f$ be a transcendental meromorphic function and $\varphi(z)(\not\equiv 0,\infty)$ be a small function of $f$. If $q_{0}(\geq 3+\mu_{\ast})$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ are integers. Then

[TABLE]

Remark 2.2**.**

Clearly, Corollary 2.1 extends Theorem F.

Corollary 2.2**.**

Let $f$ be a transcendental meromorphic function and $\varphi(z)(\not\equiv 0,\infty)$ be a small function of $f$. If $q_{0}(\geq 3+\mu_{\ast})$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ are integers. Then $M[f]-\varphi(z)$ has infinitely many zeros.

Theorem 2.2**.**

Let $f$ be a transcendental meromorphic function and $\varphi(z)(\not\equiv 0,\infty)$ be a small function of $f$. If $q_{0}(\geq 3)$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ are integers. Then

[TABLE]

Corollary 2.3**.**

Let $f$ be a transcendental meromorphic function and $\varphi(z)(\not\equiv 0,\infty)$ be a small function of $f$. If $q_{0}(\geq 3)$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ are integers. Then $M[f]-\varphi(z)$ has infinitely many zeros.

Theorem 2.3**.**

Let $f$ be a transcendental meromorphic function and $\varphi(z)(\not\equiv 0,\infty)$ be a small function of $f$. If every pole of $f$ has multiplicity atleast $l(\geq 1)$ and $q_{0}(\geq 1+\frac{2}{l})$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ are integers, then

[TABLE]

Corollary 2.4**.**

Let $f$ be a transcendental meromorphic function having no simple pole and $\varphi(z)(\not\equiv 0,\infty)$ be a small function of $f$. If $q_{0}(\geq 2)$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ are integers, then $M[f]-\varphi(z)$ has infinitely many zeros.

3. Lemmas

Let $M[f]=a(z)(f)^{q_{0}}(f^{\prime})^{q_{1}}...(f^{(k)})^{q_{k}}$ be a differential monomial generated by a transcendental meromorphic function $f$ and $a(z)$ be a small function of $f$.

In this paper, we assume that $q_{0}(\geq 1)$ and $q_{k}(\geq 1)$ and $f$ is a transcendental meromorphic function.

Lemma 3.1**.**

([17]) Let $f$ be a non-constant meromorphic function on $\mathbb{C}$, and let $a_{l},\ldots,a_{q}$ be distinct meromorphic functions on $\mathbb{C}$. Assume that $a_{i}$ are small functions with respect to $f$ for all $i=1,\ldots,q$. Then we have the second main theorem,

[TABLE]

for all $\varepsilon>0$.

Lemma 3.2**.**

For a non constant meromorphic function $g$, we obtain

[TABLE]

Proof.

The proof is same as the formula (12) of ([6]). ∎

Lemma 3.3**.**

([18]) Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$. Then

[TABLE]

Lemma 3.4**.**

Let $M[f]$ be differential polynomial generated by a transcendental meromorphic function $f$. Then $M[f]$ is non-constant.

Proof.

Here

[TABLE]

Thus by the first fundamental theorem and lemma of logarithmic derivative, we have

[TABLE]

Since $f$ is a transcendental meromorphic function, so by Lemma 3.3 and inequlity (3.1), $M[f]$ must be non-constant. ∎

Lemma 3.5**.**

Let $f$ be a transcendental meromorphic function and $M[f]$ be a differential polynomial in $f$, then

[TABLE]

Proof.

The proof is similar to the proof of the Lemma 2.4 of ([9]). ∎

4. Proof of the Theorems

Proof of Theorem 2.1.

Now

[TABLE]

Thus by the first fundamental theorem and lemma of logarithmic derivative, we have

[TABLE]

Now, by Lemma 3.1, we have

[TABLE]

for all $\varepsilon>0$.

Let $z_{0}$ be a zero of $f$ with multiplicity $q(\geq 1)$. We assume that $z_{0}$ is not a zero or pole of $\varphi(z)$. Now we consider two cases:

Case-I If $q\geq k+1$, then $z_{0}$ is a zero of $M[f]$ of order $q\mu-\mu_{\ast}$. Now

[TABLE]

Case-II If $q\leq k$, then $z_{0}$ is a zero of $M[f]$ of order $qq_{0}+(q-1)q_{1}+(q-2)q_{2}+\ldots+2q_{q-2}+q_{q-1}$. Now

[TABLE]

Now, by the first fundamental theorem, we obtain

[TABLE]

and,

[TABLE]

so from the above dicussion, we have

[TABLE]

Combining (4),(4.2) and (4), we have

[TABLE]

This completes the proof. ∎

Proof of Theorem 2.2.

Assume that $b=b(z)=\frac{1}{\varphi(z)}$. Now by Lemma 3.4, it is clear that $b(z)M[f]$ is non-constant. Again

[TABLE]

Thus in view of Lemmas 3.2 and 3.5, the first fundamental theorem and lemma of logarithmic derivative, we have

[TABLE]

Let us define $g:=bM[f]-1$ and $h:=\frac{g^{\prime}}{f^{q_{0}-1}}$. Then

[TABLE]

If $q_{0}\geq 3$, then from (4), we have

[TABLE]

∎

Proof of Theorem 2.3.

Using (4) and the first fundamental theorem, we have

[TABLE]

That is,

[TABLE]

∎

This completes the proof.

5. Applications

Let $D\subset\mathbb{\overline{C}}$ be a domain. A family $\mathscr{F}$ of analytic functions in $D$ is said to be normal if every sequence $\{f_{n}\}\subset\mathscr{F}$ has a convergent subsequence, which converges spherically, locally and uniformly in $D$ to a analytic function or $\infty$.

The aim of this sectionis to provide normality criterion for a family of analytic functions.

Using Mues’s result ([10]), Pang ([11]) proved the following result:

Theorem I**.**

([11]) Let $\mathscr{F}$ be a family of meromorphic function on a domain $D$. If each $f\in\mathscr{F}$ satisfies $f^{2}f^{\prime}\not=1$, then $\mathscr{F}$ is normal on domain $D$.

In this sequel, in 2005, Huang and Gu ([5]) proved the following theorem:

Theorem J**.**

([5]) Let $\mathscr{F}$ be a family of meromorphic functions on a domain $D$ and let $k$ be a positive integer. If for each $f\in\mathscr{F}$, $f$ has only zeros of multiplicity at least $k$ and $f^{2}f^{(k)}\not=1$, then $\mathscr{F}$ is normal on domain $D$.

Using Theorem 2.2, we provide a normality criterion for a family of analytic functions.

Theorem 5.1**.**

Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq 1)$, $q_{0}(\geq 3)$, $q_{i}(\geq 0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq 1)$ be positive integers. If for each $f\in\mathscr{F}$,

1. i.

$f$ has only zeros of multiplicity at least $k$ 2. ii.

$f^{q_{0}}(f^{\prime})^{q_{1}}\ldots(f^{(k)})^{q_{k}}\not=1$,

then $\mathscr{F}$ is normal on domain $D$.

Before going to prove the above result, we need to recall a lemma.

Lemma 5.1**.**

([12]) Let $\mathscr{F}$ be a family of meromorphic functions on the unit disc $\Delta$ such that all zeros of functions in $\mathscr{F}$ have multiplicity at least $k$. Let $\alpha$ be a real number satisfying $0\leq\alpha<k$. Then $\mathscr{F}$ is not normal in any neighbourhood of $z_{0}\in\Delta$ if and only if there exist

1. (i)

points $z_{k}\in\Delta$, $z_{k}\rightarrow z_{0}$; 2. (ii)

positive numbers $\rho_{k}$, $\rho_{k}\rightarrow 0$; and 3. (iii)

functions $f_{k}\in\mathscr{F}$

such that $\rho_{k}^{-\alpha}f_{k}(z_{k}+\rho_{k}\zeta)\rightarrow g(\zeta)$ spherically uniformly on compact subsets of $\mathbb{C}$, where $g$ is a nonconstant meromorphic function.

Proof of the Theorem 5.1.

Since normality is a local property, we may assume that $D=\Delta$. If possible, suppose that $\mathscr{F}$ is not normal on $\Delta$, then by Lemma 5.1, there exist $\{f_{n}\}\subset\mathscr{F}$, $z_{n}\in\Delta$ and positive numbers $\rho_{k}$ with $\rho_{k}\rightarrow 0$ such that

[TABLE]

locally, uniformly in spherical metric. Let

[TABLE]

and,

[TABLE]

Then

[TABLE]

Now, we made the following observations:

1. a.

by Lemma 5.1, $g(\zeta)$ is non-constant meromorphic function. 2. b.

by Hurwitz’s Theorem (pp. 152, [2]), all zeros of $g(\zeta)$ are of multiplicity atleast $k$. 3. c.

$H(\zeta)\not\equiv 0$, otherwise, $g(\zeta)$ will become a polynomial of degree atmost $k-1$, which is impossible by (b). 4. d.

by Hurwitz’s Theorem, $H(\zeta)\not=1$, as $H_{n}(\zeta)\not=1$. 5. e.

by Theorem 2.2 and (d), $g(\zeta)$ must be non-trancendental, i.e., non-constant rational function. 6. f.

Since $\mathscr{F}$ is a family of analytic functions, so $g_{n}(\zeta)$ is analytic. Since, $g_{n}(\zeta)\rightarrow g(\zeta)$ locally, uniformly in spherical metric and $g(\zeta)$ is non-constant, hence, $g(\zeta)$ is analytic.

Thus using (e) and (f), we can conclude that $g(\zeta)$ is a non-constant polynomial function, say, $g(\zeta)=c_{0}+c_{1}\zeta+\ldots+c_{l}\zeta^{l}$. But by (b), $l$ must be atleast $k$. Thus $H(\zeta)$ is a non-constant polynomial. So, by the Fundamental Theorem of Algebra, $H(\zeta)=1$ has a solution, which contradicts (d). Thus our assumption is wrong. So $\mathscr{F}$ is normal. This completes the proof. ∎

Proceeding as above and using the result of Karmakar and Sahoo (Theorem 1.1 of [7]), the following result is obvious.

Corollary 5.1**.**

Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq 1)$ and $n(\geq 2)$ be two positive integers. If for each $f\in\mathscr{F}$,

1. i.

$f$ has only zeros of multiplicity at least $k$ 2. ii.

$f^{n}f^{(k)}\not=1$,

then $\mathscr{F}$ is normal on domain $D$.

Acknowledgement

We are thankful to Prof. C. C. Yang for giving us some ardent help and suggestion in the time of the preparation of this manuscript.