Normal Criterion for Families of Meromorphic Functions and Shared Functions

Abstract

This paper broadens the scope of existing research: the shared value is generalized from a non-zero finite complex number to a non-identically zero holomorphic function, the order of the derivative is extended from the first order to an arbitrary k-th order, and the constraint condition on the polynomial H is simplified to $degH\ge 2$. A more general normality criterion for families of meromorphic functions involving the sharing of differential polynomials is proved. Let D be a domain, $\mathcal{F}$ be a family of meromorphic functions in D, and $P\left(z\right)$ be a non-identically zero holomorphic function in D. If for any $f,g\in \mathcal{F}$, the differential polynomials $H\left(f\right)f^{\left(k\right)}$ and $H\left(g\right)g^{\left(k\right)}$ share $P\left(z\right)$ in D, then $\mathcal{F}$ is normal in D.

1. Introduction and Preliminaries

1.1. Preliminaries

This section briefly outlines the core fundamentals of the normality theory of meromorphic functions and elaborates on the key theorems of Nevanlinna theory. All relevant notations and expressions involved in this paper adhere to the standard conventions established in reference [1].

As a natural generalization of holomorphic functions, meromorphic functions allow for the existence of singularities such as poles, serving as the core research objects in the theories of value distribution and normal families within the field of complex analysis. The function families discussed in this paper are all families of single-valued analytic functions containing only poles as singularities (i.e., meromorphic function families), which will not be repeatedly specified in subsequent discussions.

For a sequence of meromorphic functions $\left\{f_n\right\}$ and a meromorphic function f in a domain D, if $\left\{f_n\right\}$ converges uniformly to f with respect to the spherical metric on every compact subset of D, this is denoted as $f_n\stackrel{\chi }{\Rightarrow }f$ (in D); if the convergence is with respect to the Euclidean metric, it is denoted as $f_n\to f$ (in D).

Let f and g be meromorphic functions in a domain D and let a be a complex constant. If $f\left(z\right)-a$ and $g\left(z\right)-a$ have the same zeros in D Ignoring Multiplicities (IM), then f and g are said to share the value a IM in D; if the zeros are identical and their multiplicities are completely consistent, this is called Counting Multiplicities (CM) sharing. All sharing relations involved in this paper are IM sharing, which will not be separately stated hereafter.

A family $\mathcal{F}$ of meromorphic functions on a subset D of the complex plane $\mathbb{C}$ is said to be normal on D if every sequence $\left\{f_n\right\}$ in $\mathcal{F}$ has a subsequence $\left\{f_{n_j}\right\}$ that converges locally uniformly to a function f on D with respect to the spherical metric (the limit function f may take the value ∞). The core characteristic of a normal family is its “sequential compactness”, meaning that every sequence has a convergent subsequence.

For a meromorphic function $f\left(z\right)$, $n(r,f)$ denotes the number of zeros of $f\left(z\right)$ in the closed disk $\left|z\right|\le r$, with the counting function $N(r,f)$ defined as

$$N(r,f)={\int }_0^r\frac{n(t,f)-n(0,f)}{t}dt+n(0,f)logr.$$

Correspondingly, the proximity function $m(r,f)$ and characteristic function $T(r,f)$ are respectively defined as

$$m(r,f)=\frac{1}{2\pi }{\int }_0^{2\pi }log\frac{1}{\left|f\right(r{e}^{i\theta }\left)\right|}d\theta ,T(r,f)=m(r,f)+N(r,f).$$

The above three functions are the core tools of Nevanlinna theory: $n(r,f)$ realizes the local counting of zeros, and $N(r,f)$ extends the local counting to the global average distribution frequency through integration; $m(r,f)$ characterizes the average degree to which the function approaches the value a on the circle $\left|z\right|=r$; finally, the characteristic function $T(r,f)$ comprehensively reflects the growth and value distribution properties of the meromorphic function, and serves as a key bridge connecting the growth behavior and value distribution law of the function.

Nevanlinna theory constitutes the core of the value distribution theory of meromorphic functions. Its First Fundamental Theorem establishes the fundamental connection between the characteristic function and the distribution of value points, while its Second Fundamental Theorem reveals the intrinsic restrictive relationship among the frequencies at which a meromorphic function takes multiple distinct values. Together, these two theorems form an important mathematical tool for researching meromorphic functions.

Nevanlinna’s First Fundamental Theorem: Let $f\left(z\right)$ be a non-constant meromorphic function in $\left|z\right|<R$ ($0<R\le \infty$) and let a be any finite complex number; then, for $0<r<R$, the following holds:

$$T\left(r,\frac{1}{f-a}\right)=T(r,f)+h(a,r),$$

where

$$\left|h(a,r)\right|\le |log|c_s\left|\right|+{log}^{+}\left|a\right|+log2$$

and $f\left(z\right)-a=c_s{z}^s+c_{s+1}{z}^{s+1}+\dots (c_s\ne 0)$.

This theorem indicates that the characteristic function $T(r,f)$ essentially characterizes the average frequency at which the meromorphic function $f\left(z\right)$ takes any finite complex value a, and this characterization differs only by a constant term for different values of a.

Nevanlinna’s Second Fundamental Theorem: Let $f\left(z\right)$ be a non-constant meromorphic function in $\left|z\right|<R$ ($0<R\le \infty$) and let $a_j$ ($j=1,2,\dots ,q$) be $q(\ge 3)$ distinct complex numbers (finite or infinite) satisfying $f\left(0\right)\ne 0,\infty$ and $f^{\prime }\left(0\right)\ne 0$; then,

$$(q-2)T(r,f)\le \sum _{j=1}^{q}N\left(r,\frac{1}{f-a_j}\right)-N_1\left(r\right)+S(r,f),$$

where

$$N_1\left(r\right)=2N(r,f)-N(r,f^{\prime })+N\left(r,\frac{1}{f^{\prime }}\right)$$

and where $S(r,f)=o\left(T\right(r,f\left)\right)$ (as $r\to \infty$) denotes the error term.

This theorem uncovers the intrinsic constraint on the value distribution of meromorphic functions: there exists a profound correlation among the frequencies at which a meromorphic function takes multiple distinct values, and the sum of the counting functions of all its value points cannot all be significantly smaller than the characteristic function $T(r,f)$.

1.2. Introduction

There exists a close connection between normal families and normal functions which justifies the reasonable expectation of inherent consistency between the criteria for determining normal functions and the established criteria for normal families. The French mathematician Montel first proposed the concept of normal families of meromorphic functions. Since its establishment in 1912, this concept has occupied an important position and played a crucial role in the field of complex analysis. From the 1920s to the 1950s, research on the normality of function families focused on holomorphic function families. Between the 1950s and the 1980s, Hayman proposed a series of conjectures concerning the normality of meromorphic function families, but their proof methods are highly complex, requiring the use of Nevanlinna theory to establish inequalities related to characteristic functions followed by completing the proof through eliminating the original values. This process demands extremely high technical skills, leading to a significant increase in the difficulty of proving some normality criteria. In 1975, the Israeli mathematician Zalcman broke through the limitations of traditional methods and established the necessary and sufficient conditions for the non-normality of meromorphic function families, providing a brand-new perspective for research in this field. Since then, research on the normality criteria of meromorphic function families has entered a stage of rapid development.

In 1992, Schwick first revealed the intrinsic connection between the normality of function families and shared values. By virtue of Zalcman’s lemma and Nevanlinna theory, he established a normality criterion for function families sharing three values in ref. [2]. Subsequently, Pang and Zalcman in ref. [3] and Chang, Fang et al. in ref. [4] reduced the number of shared values for this problem from three to two. In 2004, Fang and Zalcman proved in ref. [5] that if $\mathcal{F}$ is a family of meromorphic functions in a domain D, and for any two functions $f_n\left(z\right),g_n\left(z\right)\in \mathcal{F}$ it is the case that they share 0 and $f_n^l\left(z\right)f_n^{\prime }\left(z\right)$ shares a non-zero constant a with $g_n^l\left(z\right)g_n^{\prime }\left(z\right)$, then $\mathcal{F}$ is normal in D.

In 2008, Zhang made an important improvement to the results of [5] in ref. [6]: when $l\ge 2$, a more refined form of conclusion can be obtained by weakening the sharing condition (removing the requirement of sharing the value 0). In 2010, Lei et al. in ref. [7] conducted an in-depth analysis on the basis of existing research and generalized the differential monomial $f_n^l\left(z\right)f_n^{\prime }\left(z\right)$ in ref. [5,6] to a differential polynomial of the form ${\sum }_{t=3}^l{a}_t{f}_n^t\left(z\right)f_n^{\prime }\left(z\right)$, obtaining the following theorem.

Theorem 1. 

Let $\mathcal{F}$ be a family of meromorphic functions in a domain D and let H be a polynomial satisfying $degH\ge 3$, or $degH=2$ with only one zero. Let b be a non-zero finite complex number. If for any two functions $f_n\left(z\right),g_n\left(z\right)\in \mathcal{F}$, $H\left(f_n\left(z\right)\right)f_n^{\prime }\left(z\right)$ shares b with $H\left(g_n\left(z\right)\right)g_n^{\prime }\left(z\right)$, then the function family $\mathcal{F}$ is normal in D.

In 2017, Deng et al. conducted an in-depth study on the conclusions of [6] in ref. [8] and successfully generalized their results on constants to the class of holomorphic functions, obtaining the following more general theorem.

Theorem 2. 

Given three integers $k\ge 1$, $l\ge 2$, $m\ge 0$, let $\mathcal{F}$ stand for a family of meromorphic functions within a domain D. It is required that for every $f\in \mathcal{F}$, each zero of f possesses a multiplicity of at least $k+m$. Let $h\left(z\right)$ (a non-identically zero holomorphic function on D) satisfy the requirement that the multiplicity of each of its zeros does not exceed m. If for any pair of functions $f,g\in \mathcal{F}$, $f^l\left(z\right)f^{\left(k\right)}\left(z\right)-h\left(z\right)$ has no more than one distinct zero in D, then $\mathcal{F}$ is normal in D.

In recent years, research on the normality of meromorphic function families has attracted widespread attention from scholars, particularly focusing on normality criteria involving shared values (or shared functions). Numerous scholars have conducted extensive investigations into the normality problems related to shared values within the same function family and achieved a series of valuable results (see [1,9,10,11,12,13,14]). Additionally, some scholars have turned their attention to the sharing problems between different function families; relevant research findings can be found in ref. [15,16,17,18,19,20].

Despite the rich theoretical system established in the existing literature, there remain several aspects worthy of in-depth exploration. First, Theorem 1 only discusses the first-order differential polynomial $H\left(f_n\left(z\right)\right)f_n^{\prime }\left(z\right)$, and there is a lack of systematic research on the case of arbitrary higher-order (k-th order) differential polynomials $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)$. The shared value is limited to a non-zero constant b, while the more complex scenario of sharing a holomorphic function $P\left(z\right)$ is common in practical applications. Whether extending the constant to a function will change the essential conclusion of the theorem remains an issue to be verified. Second, it is unclear whether the structure of the differential polynomial is optimal (i.e., whether the condition $degH\ge 3$, or $degH=2$ with only one zero can be weakened).

Based on the above considerations, this paper aims to relax the restrictive conditions of existing theorems, introduce more general differential polynomials and shared functions, and explore the results obtained while maintaining the core logical framework of the theorems.

2. Main Results

This paper follows the proof strategy of Theorem 2 to further generalize and improve Theorem 1. While retaining the core framework of Theorem 1, we extend the differential polynomial from the first-order form $H\left(f\right)f^{\prime }$ to an arbitrary k-th order form $H\left(f\right)f^{\left(k\right)}$, thereby broadening the applicability of the theorem from specific first-order derivatives to higher-order derivatives. The shared object is refined from a constant b to an arbitrary non-identically zero holomorphic function $P\left(z\right)$, and restrictions on the multiplicity of its zeros are considered (bounded above by m). Correspondingly, the condition on the multiplicity of zeros of functions in the family $\mathcal{F}$ is adjusted to be at least $k+m$. Furthermore, the structural condition on the differential polynomial is optimized to be sharp; specifically, the restriction that $degH\ge 3$ or $degH=2$ with only one zero is weakened to $degH\ge 2$, yielding a more general conclusion.

For convenience of subsequent exposition, differential polynomials of the form ${\sum }_{t=2}^l{a}_t{f}_n^t\left(z\right)f_n^{\left(k\right)}\left(z\right)$ that appear repeatedly throughout this paper are uniformly denoted as $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)$, where H is a polynomial of degree at least 2.

The main results of this paper are stated below.

Theorem 3. 

Let k and m be positive integers satisfying $k\ge 1$ and $m\ge 1$, and let $\mathcal{F}$ be a family of meromorphic functions in a domain D. Each function in $\mathcal{F}$ has zeros with multiplicity at least $k+m$. Let $P\left(z\right)$ be a non-identically zero holomorphic function in D with zeros that have multiplicity at most m. If for any two functions $f,g\in \mathcal{F}$, $H\left(f\right)f^{\left(k\right)}$ and $H\left(g\right)g^{\left(k\right)}$ share $P\left(z\right)$ in D, then $\mathcal{F}$ is normal in D.

Corollary 1. 

Let k be a positive integer ($k\ge 1$) and a be a finite complex number. Let $\mathcal{F}$ be a family of meromorphic functions on a domain D. For any $f\in \mathcal{F}$, suppose the multiplicity of zeros of f is at least $k+1$ and the multiplicity of poles of f is at least 2. Additionally, for any two functions $f,g\in \mathcal{F}$, if $H\left(f\right)f^{\left(k\right)}$ and $H\left(g\right)g^{\left(k\right)}$ share a in D, then $\mathcal{F}$ is normal in D.

Through in-depth research, this paper further extends the above results and obtains the following theorem.

Theorem 4. 

Let $\mathcal{F}$ be a family of meromorphic functions on a domain D and let $k,m$ be positive integers with $k\ge 1$ and $m\ge 1$. Let $P\left(z\right)$ be a non-identically zero holomorphic function on D, where the multiplicity of each zero of $P\left(z\right)$ is at most m. For any $f\in \mathcal{F}$, if the multiplicity of zeros of f is at least $k+m$, the multiplicity of poles of f is at least 2, and the differential polynomial $H\left(f\right)f^{\left(k\right)}-P\left(z\right)$ has at most one zero in D, then $\mathcal{F}$ is normal in D.

Corollary 2. 

Let k be a positive integer with $k\ge 1$ and let a be a finite complex number. Let $\mathcal{F}$ be a family of meromorphic functions on a domain D. For any $f\in \mathcal{F}$, if the multiplicity of zeros of f is at least $k+1$, the multiplicity of poles of f is at least 2, and the differential polynomial $H\left(f\right)f^{\left(k\right)}-a$ has at most one distinct zero in D, then $\mathcal{F}$ is normal in D.

In the following, we will systematically explain the necessity of each condition in Theorem 4 by constructing specific counterexamples. These counterexamples will verify one-by-one that when any of the conditions (such as $degH$, the multiplicity limit of zeros of $P\left(z\right)$, the requirements on the multiplicities of zeros and poles of functions, and the constraint on the number of zeros of differential polynomials) is not satisfied, then the conclusion of the theorem may not hold, thereby fully demonstrating the rigor and indispensability of the setting of each condition.

Example 1. 

Let $\Delta =\{z:|z|<1\}$ be the unit disk in the complex plane; in addition, define the function family $\mathcal{F}={\left\{f_n\right\}}_{n=1}^{\infty }$, where for any positive integer n, $f_n\left(z\right)$ is defined as $f_n\left(z\right)=n{z}^2+\frac{1}{2n}$. Take the polynomial $H\left(z\right)=z$ and the holomorphic function $P\left(z\right)=1$ (a constant function).

Direct calculation gives $H\left(f_n\left(z\right)\right)f_n^{″}\left(z\right)=f_n\left(z\right)f_n^{″}\left(z\right)=2n^2{z}^2+1$. It follows that for any positive integers $m,n$, $H\left(f_m\left(z\right)\right)f_m^{″}\left(z\right)$ shares the holomorphic function $P\left(z\right)=1$ with $H\left(f_n\left(z\right)\right)f_n^{″}\left(z\right)$. According to the definition of normal families, the function family $\mathcal{F}$ does not satisfy equicontinuity at $z=0$; thus, $\mathcal{F}$ is not normal in $\Delta$. This result shows that when the degree of the polynomial $H\left(z\right)$ is $degH=1$, which does not satisfy the condition $degH\ge 2$ in Theorem 4, the conclusion of the theorem does not hold, highlighting the necessity of the condition $degH\ge 2$.

Example 2. 

Let $\Delta =\{z:|z|<1\}$ be the unit disk; additionally, define the function family $\mathcal{F}={\left\{f_n\right\}}_{n=1}^{\infty }$, where $f_n\left(z\right)=e^{nz}$ for any positive integer n. Take the polynomial $H\left(z\right)=z^q$ ($q\ge 2$ is an integer) and the holomorphic function $P\left(z\right)\equiv 0$ (the identically zero function).

For any positive integer n and $k\ge 1$ given in Theorem 4, direct calculation provides $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)-P\left(z\right)={\left(e^{nz}\right)}^q·n^k{e}^{nz}=n^k{e}^{(qn+k)z}$, which has no zeros in $\Delta$. However, according to the normality criterion of families, the function family $\mathcal{F}$ is not normal at $z=0$. This result directly shows that when $P\left(z\right)\equiv 0$, which does not satisfy the condition $P\left(z\right)\not\equiv 0$ in Theorem 4, the conclusion of the theorem no longer holds, fully demonstrating the indispensability of $P\left(z\right)\not\equiv 0$.

Example 3. 

Let $\Delta =\{z:|z|<1\}$ be the unit disk in the complex plane and define the function family $\mathcal{F}={\left\{f_n\right\}}_{n=1}^{\infty }$, where for any positive integer n, $f_n\left(z\right)=n{z}^{k-1}$ ($k\ge 1$ is the integer given in Theorem 4). Take the holomorphic function $P\left(z\right)=1$ (a constant function) and the polynomial $H\left(z\right)=z^q$ ($q\ge 2$ is an integer).

For any positive integer n, calculation provides $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)-P\left(z\right)={\left(n{z}^{k-1}\right)}^q·n·(k-1)!=n^{q+1}(k-1)!z^{q(k-1)}$, which has no zeros in $\Delta$. However, the function family $\mathcal{F}$ does not satisfy the normality condition at $z=0$, so $\mathcal{F}$ is not normal in $\Delta$. This result shows that the condition “the multiplicity of zeros of each function in $\mathcal{F}$ is at least $k+m$” in Theorem 4 cannot be weakened arbitrarily. When the multiplicity of zeros of the function is only $k-1<k+m$, the conclusion of the theorem does not hold, thereby verifying the rigor of this condition.

Example 4. 

Let $\Delta =\{z:|z|<1\}$ be the unit disk and define the function family $\mathcal{F}={\left\{f_n\right\}}_{n=1}^{\infty }$, where $f_n\left(z\right)=\frac{1}{nz}$ for any positive integer n. Take the polynomial $H\left(z\right)=z^q$ ($q\ge 2$ is an integer) and the meromorphic function $h\left(z\right)=\frac{1}{z^{q+3}}$, and let $P\left(z\right)=z^q$ ($q\ge 2$ is an integer).

Direct calculation gives $H\left(f_n\left(z\right)\right)f_n^{″}\left(z\right)-h\left(z\right)={\left(\frac{1}{nz}\right)}^q·\frac{2}{n{z}^3}-\frac{1}{z^{q+3}}=\frac{2-n^{q+1}}{n^{q+1}{z}^{q+3}}$, which has no zeros in $\Delta$. However, the function family $\mathcal{F}$ is not normal at $z=0$, so $\mathcal{F}$ is not normal in $\Delta$. This result shows that the condition “the shared object is a holomorphic function $P\left(z\right)$” in Theorem 4 cannot be replaced by a meromorphic function. When the shared object is a meromorphic function $h\left(z\right)$, the conclusion of the theorem does not hold, highlighting the necessity of the condition that $P\left(z\right)$ be a holomorphic function.

3. Some Lemmas

Lemma 1 

([21]). Let $\mathcal{F}$ be a family of meromorphic functions in the unit disk Δ and let the multiplicity of zeros of each function in $\mathcal{F}$ be no less than k. Let α be a real number satisfying $0\le \alpha \le k$. If $\mathcal{F}$ is not normal at $z_0\in \Delta$, then the following exist:

(1) A sequence of points $z_n\in \Delta$ such that $z_n\to z_0$

(2) A sequence of functions $\left\{f_n\right\}\subseteq \mathcal{F}$

(3) A sequence of positive numbers ${\rho }_n\to 0$

such that $g_n\left(\zeta \right)={\rho }_n^{-\alpha }{f}_n(z_n+{\rho }_n\zeta )\stackrel{\chi }{\Rightarrow }g\left(\zeta \right)$, where g is a non-constant meromorphic function on the complex plane $\mathbb{C}$, the multiplicity of zeros of g is at least k, and its spherical derivative satisfies $g^{\#}\left(\zeta \right)\le g^{\#}\left(0\right)=kA+1$.

Lemma 2 was not independently completed by a single scholar, but was gradually improved by multiple scholars in different periods. In this paper, we cite the form given by Deng et al. in ref. [8].

Lemma 2. 

Let $l\ge 2$ and $k\ge 1$ be positive integers and let a be a non-zero finite complex constant. If $f\left(z\right)$ is a non-constant meromorphic function with zeros of multiplicity at least k, then $f^l\left(z\right)f^{\left(k\right)}\left(z\right)-a$ has at least two distinct zeros in the complex plane $\mathbb{C}$.

Lemma 3 

([8]). Let $m,k\ge 1$ and $l\ge 2$ be positive integers and let $H\left(z\right)$ be a polynomial of degree m. If $f\left(z\right)$ is a non-constant meromorphic function with zeros of multiplicity at least $k+m$, then $f^l\left(z\right)f^{\left(k\right)}\left(z\right)-H\left(z\right)$ has at least two distinct zeros in the complex plane $\mathbb{C}$.

Lemma 4. 

Let $k\ge 1$ be a positive integer, $P\left(z\right)$ be a holomorphic function in D that is not identically zero, and $\mathcal{F}$ be a family of meromorphic functions in D. If for any $f\in \mathcal{F}$ the multiplicity of zeros of $f\left(z\right)$ is at least $k+1$, the multiplicity of poles of $f\left(z\right)$ is at least 2, H has at least two distinct zeros, and the differential polynomial $H\left(f\right)f^{\left(k\right)}-P\left(z\right)$ has at most one zero in D, then $\mathcal{F}$ is normal in D.

Proof. 

According to the conditions of the lemma, we make the following assumption:

$$H\left(z\right)=R\left(z\right)(z-a)(z-b),$$

(1)

where $R\left(z\right)$ is a non-zero polynomial and where $a,b$ are two distinct finite complex numbers in $\mathbb{C}$. Assume that the function family $\mathcal{F}$ is not normal at a point $z_0$ in the domain D. According to Lemma 1, we have

$${\psi }_n\left(\zeta \right)=f_n(z_n+{\rho }_n\zeta )\stackrel{\chi }{\Rightarrow }\psi \left(\zeta \right).$$

(2)

Here, the sequence of functions $f_n$, sequence of points $z_n$, and sequence of positive numbers ${\rho }_n$ possess the corresponding properties as in Lemma 1, and Equation (2) holds uniformly with respect to the spherical distance on each compact subset of the complex plane $\mathbb{C}$. Moreover, $\psi \left(\zeta \right)$ is a non-constant meromorphic function in $\mathbb{C}$, where the multiplicity of its zeros is at least $k+1$ and the multiplicity of its poles is at least 2.

On any compact subset of the complex plane $\mathbb{C}$ that does not contain the poles of $\psi \left(\zeta \right)$, the following result can be obtained from Equation (2):

$${\rho }_n^{-i}{\psi }_n^{\left(i\right)}\left(\zeta \right)=f_n^{\left(i\right)}(z_n+{\rho }_n\zeta ),(i=1,2,\cdots ,k).$$

(3)

Combining Equations (1)–(3), we obtain

$$\begin{array}{cc}\hfill & {\rho }_n^k\left[H\left(f_n(z_n+{\rho }_n\zeta )\right)f_n^{\left(k\right)}(z_n+{\rho }_n\zeta )-P(z_n+{\rho }_n\zeta )\right]\hfill \\ \hfill & =H\left({\psi }_n\left(\zeta \right)\right){\psi }_n^{\left(k\right)}\left(\zeta \right)-{\rho }_n^{k}P(z_n+{\rho }_n\zeta )\hfill \\ \hfill & \to R\left(\psi \left(\zeta \right)\right)(\psi \left(\zeta \right)-a)(\psi \left(\zeta \right)-b){\psi }^{\left(k\right)}\left(\zeta \right).\hfill \end{array}$$

(4)

Claims:

(1) Equation (4) is not identically zero.

(2) The number of distinct zeros of Equation (4) in the complex plane $\mathbb{C}$ is at most 1.

For Claim 1, assume that Equation (4) is identically zero; then, the function $g\left(\zeta \right)$ is a polynomial of degree at most k, which is a contradiction. Thus, Claim 1 is proved.

For Claim 2, we first verify the existence of zeros of Equation (4). Suppose that Equation (4) has no zeros, that is, $R\left(\psi \left(\zeta \right)\right)(\psi \left(\zeta \right)-a)(\psi \left(\zeta \right)-b){\psi }^{\left(k\right)}\left(\zeta \right)\ne 0$. It follows that $\psi \left(\zeta \right)$ never takes the values a or b, and its k-th derivative ${\psi }^{\left(k\right)}\left(\zeta \right)$ never takes the value 0. Together with the fact that the multiplicity of its zeros is at least $k+1$ and the multiplicity of its poles is at least 2, $\psi \left(\zeta \right)$ must be a constant function, which is a contradiction. Therefore, Equation (4) has at least one zero on the complex plane $\mathbb{C}$.

Next, assume that Equation (4) has two distinct zeros, ${\zeta }_1$ and ${\zeta }_2$. According to the isolation of zeros of meromorphic functions, we can take a sufficiently small positive number $\delta >0$ such that the two neighborhoods $D_1=\{\zeta :|\zeta -{\zeta }_1|<\delta \}$ and $D_2=\{\zeta :|\zeta -{\zeta }_2|<\delta \}$ contain only the zeros ${\zeta }_1$ and ${\zeta }_2$, respectively, and $D_1\cap D_2=⌀$. Meanwhile, in the union region $D_1\cup D_2$, Equation (4) has no other zeros and no poles except the two known zeros ${\zeta }_1$ and ${\zeta }_2$. According to Hurwitz’s theorem, when n is sufficiently large, there exist points ${\zeta }_{1n}\in D_1$ and ${\zeta }_{2n}\in D_2$ such that ${\zeta }_{1n}\to {\zeta }_1$ and ${\zeta }_{2n}\to {\zeta }_2$ as $n\to \infty$. Thus, we have

$$H\left(f_n(z_n+{\rho }_n{\zeta }_{1n})\right)f_n^{\left(k\right)}(z_n+{\rho }_n{\zeta }_{1n})-P(z_n+{\rho }_n{\zeta }_{1n})=0,$$

$$H\left(f_n(z_n+{\rho }_n{\zeta }_{2n})\right)f_n^{\left(k\right)}(z_n+{\rho }_n{\zeta }_{2n})-P(z_n+{\rho }_n{\zeta }_{2n})=0.$$

According to the assumption of Lemma 4, $H\left(f\left(z\right)\right)f^{\left(k\right)}\left(z\right)-P\left(z\right)$ has at most one zero, so

$$z_n+{\rho }_n{\zeta }_{1n}=z_n+{\rho }_n{\zeta }_{2n}={\zeta }_0.$$

This implies that ${\zeta }_{1n}={\zeta }_{2n}=\frac{{\zeta }_0-z_n}{{\rho }_n}$, which contradicts $D_1\cap D_2=⌀$. Thus, Claim 2 holds.

Therefore, Equation (4) has at most one zero. It can be inferred that either $g\left(\zeta \right)\ne a$ or $g\left(\zeta \right)\ne b$. Without loss of generality, assume that $g\left(\zeta \right)\ne a$. Combining the First and Second Fundamental Theorems of Nevanlinna and considering that the poles and zeros of $\psi \left(\zeta \right)$ are all multiple, we obtain

$$\begin{array}{cc}\hfill T(r,\psi )& <\overline{N}(r,\psi )+\overline{N}\left(r,\frac{1}{\psi -a}\right)+\overline{N}\left(r,\frac{1}{\psi -b}\right)+S(r,\psi )\hfill \\ \hfill & \le \overline{N}(r,\psi )+logr+S(r,\psi )\hfill \\ \hfill & \le \frac{1}{2}N(r,\psi )+logr+S(r,\psi )\hfill \\ \hfill & \le 2logr+S(r,\psi ).\hfill \end{array}$$

This indicates that $\psi \left(\zeta \right)$ cannot be a transcendental meromorphic function. On the other hand, since $\psi \left(\zeta \right)$ does not take at least one value in the set $\{a,b\}$, it cannot be a polynomial function either. Combining the above two points, it can be concluded that $\psi \left(\zeta \right)$ must be a non-polynomial rational function. In addition, considering that the poles of the meromorphic function $\psi \left(\zeta \right)$ have multiplicities at least 2, we can set its function form as $\psi \left(\zeta \right)=a+\frac{A}{{(\zeta -\beta )}^2}$ (where $A\ne 0$). It has exactly two distinct zeros $\zeta =\beta \pm \sqrt{\frac{A}{b-a}}$, which contradicts the conclusion obtained earlier that Equation (4) has at most one zero. Therefore, it can be concluded that the family of meromorphic functions $\mathcal{F}$ is normal in the domain D, and the proof of Lemma 4 is completed.

Lemma 5. 

Let $k\ge 1$ be an integer, $P\left(z\right)$ be a zero-free holomorphic function in a domain D, and $\mathcal{F}$ be a family of meromorphic functions in D. If for any $f\in \mathcal{F}$ the multiplicity of zeros of $f\left(z\right)$ is at least $k+1$, all zeros of H are multiple zeros, and the differential polynomial $H\left(f\right)f^{\left(k\right)}-P\left(z\right)$ has at most one zero in D, then $\mathcal{F}$ is normal in D.

Proof. 

According to the conditions of the lemma, we can set

$$H\left(z\right)=a_p{(z-b)}^p+\cdots +a_q{(z-b)}^q,(2\le q\le p)$$

(5)

where $p,q\ge 2$ are two positive integers, the coefficient $a_q\ne 0$, and b is a finite complex number. Assume that the function family $\mathcal{F}$ is not normal at the point $z_0$. Because $P\left(z\right)\ne 0$, there exists a positive number $\delta$ such that $P\left(z\right)$ is always non-zero in the disk $D_{\left(\delta \right)}=\{z:|z-z_0|<\delta \}$. For the sake of simplifying the proof, we construct a new family of meromorphic functions $\mathcal{G}=\left\{G\right|G=f-b,f\in \mathcal{F}\}$. Obviously, $\mathcal{G}$ is also not normal at $z_0$. According to the above conditions, applying Lemma 1, we obtain

$${\psi }_n\left(\zeta \right)=\frac{G_n(z_n+{\rho }_n\zeta )}{{\rho }_n^{\frac{k}{q+1}}}\stackrel{\chi }{\Rightarrow }\psi \left(\zeta \right).$$

(6)

Here, the sequence of functions $f_n$, the sequence of points $z_n$, and the sequence of positive numbers ${\rho }_n$ have the corresponding properties in Lemma 1, and Equation (6) holds uniformly on each compact subset of the complex plane $\mathbb{C}$ in terms of spherical distance. Because $\psi \left(\zeta \right)$ is a non-constant meromorphic function in $\mathbb{C}$, the multiplicity of its zeros is at least $k+1$.

On any compact subset of the complex plane $\mathbb{C}$ that does not contain the poles of $\psi \left(\zeta \right)$, the following results can be obtained from Equation (6):

$$\begin{array}{c}\hfill {\rho }_n^{\frac{k}{q+1}s}{\psi }_n^s\left(\zeta \right)=G_n^s(z_n+{\rho }_n\zeta ),(q\le s\le k),\end{array}$$

(7)

$$\begin{array}{c}\hfill {\rho }_n^{\frac{k}{q+1}-i}{\psi }_n^{\left(i\right)}\left(\zeta \right)=G_n^{\left(i\right)}(z_n+{\rho }_n\zeta ),(i=1,2,\cdots ,k).\end{array}$$

(8)

Combining Equations (5)–(8), we get

$$\begin{array}{cc}\hfill & H\left(f_n(z_n+{\rho }_n\zeta )\right)f_n^{\left(k\right)}(z_n+{\rho }_n\zeta )-P(z_n+{\rho }_n\zeta )\hfill \\ \hfill & =[a_p{(f_n(z_n+{\rho }_n\zeta )-b)}^p+\cdots +a_q{(f_n(z_n+{\rho }_n\zeta )-b)}^q]f_n^{\left(k\right)}(z_n+{\rho }_n\zeta )-P(z_n+{\rho }_n\zeta )\hfill \\ \hfill & =[a_p{G}_n^p(z_n+{\rho }_n\zeta )+\cdots +a_q{G}_n^q(z_n+{\rho }_n\zeta )]G_n^{\left(k\right)}(z_n+{\rho }_n\zeta )-P(z_n+{\rho }_n\zeta )\hfill \\ \hfill & =[a_p{\rho }_n^{\frac{kp}{q+1}}{\psi }_n^p\left(\zeta \right)+\cdots +a_q{\rho }_n^{\frac{kq}{q+1}}{\psi }_n^q\left(\zeta \right)]{\rho }_n^{\frac{k}{q+1}-k}{\psi }_n^{\left(k\right)}\left(\zeta \right)-P(z_n+{\rho }_n\zeta )\hfill \\ \hfill & \to a_q{\psi }^q\left(\zeta \right){\psi }^{\left(k\right)}\left(\zeta \right)-P\left(z_0\right).\hfill \end{array}$$

(9)

Claims:

(1) Equation (9) is not identically zero.

(2) The number of distinct zeros of Equation (9) in the complex plane $\mathbb{C}$ is at most 1.

For Claim 1, if Equation (9) is identically zero, it can be inferred that the meromorphic function $\psi \left(\zeta \right)$ has no zeros or poles in the complex plane. Because $\psi \left(\zeta \right)$ is a non-constant meromorphic function of order at most 2, there exists a finite complex number $c_1$ and a pair of non-zero finite complex numbers $(c_2,c_3)\ne (0,0)$ such that $\psi \left(\zeta \right)$ can be expressed in the following form:

$$\psi \left(\zeta \right)=e^{(c_1+c_2\zeta +c_3{\zeta }^2)}.$$

Therefore:

$$\begin{array}{cc}\hfill & a_q{\psi }^q\left(\zeta \right){\psi }^{\left(k\right)}\left(\zeta \right)-P\left(z_0\right)\hfill \\ \hfill & =a_q{e}^{q(c_1+c_2\zeta +c_3{\zeta }^2)}{\left(e^{c_1+c_2\zeta +c_3{\zeta }^2}\right)}^{\left(k\right)}-P\left(z_0\right)\hfill \\ \hfill & =e^{(q+1)(c_1+c_2\zeta +c_3{\zeta }^2)}{R}_1\left(\zeta \right)(c_2+2c_3\zeta )-P\left(z_0\right),\hfill \end{array}$$

where $R_1\left(\zeta \right)$ is a polynomial that is not identically zero, which leads to a contradiction. Thus, Claim 1 holds.

For Claim 2, assume that Equation (9) has two distinct zeros ${\zeta }_1$ and ${\zeta }_2$. According to the isolation of zeros of meromorphic functions, we can take a sufficiently small positive number $\delta$ such that the two neighborhoods $D_1=\{\zeta :|\zeta -{\zeta }_1|<\delta \}$ and $D_2=\{\zeta :|\zeta -{\zeta }_2|<\delta \}$ contain only the zeros ${\zeta }_1$ and ${\zeta }_2$ respectively, and $D_1\cap D_2=⌀$. Meanwhile, in the union region $D_1\cup D_2$, Equation (9) has no other zeros or poles except the two known zeros ${\zeta }_1$ and ${\zeta }_2$.

According to Hurwitz’s theorem, when n is sufficiently large, there exist points ${\zeta }_{1n}\in D_1$ and ${\zeta }_{2n}\in D_2$ such that ${\zeta }_{1n}\to {\zeta }_1$ and ${\zeta }_{2n}\to {\zeta }_2$ as $n\to \infty$. Thus, we have:

$$H\left(f_n(z_n+{\rho }_n{\zeta }_{1n})\right)f_n^{\left(k\right)}(z_n+{\rho }_n{\zeta }_{1n})-P(z_n+{\rho }_n{\zeta }_{1n})=0,$$

$$H\left(f_n(z_n+{\rho }_n{\zeta }_{2n})\right)f_n^{\left(k\right)}(z_n+{\rho }_n{\zeta }_{2n})-P(z_n+{\rho }_n{\zeta }_{2n})=0.$$

According to the assumption of Lemma 5, $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)-P\left(z\right)$ has at most one zero ${\zeta }_0$ in D; thus,

$$z_n+{\rho }_n{\zeta }_{1n}=z_n+{\rho }_n{\zeta }_{2n}={\zeta }_0.$$

Put another way, ${\zeta }_{1n}={\zeta }_{2n}=\frac{{\zeta }_0-z_n}{{\rho }_n}$ (where ${\zeta }_0$ is the zero of $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)-P\left(z\right)$), which contradicts $D_1\cap D_2=⌀$. Thus, Claim 2 holds. On the contrary, the conclusion of the lemma indicates that Equation (9) has at least two distinct zeros, which contradicts the above derivation. Therefore, it can be concluded that the function family $\mathcal{F}$ is normal in the domain D, and the proof of the lemma is completed.

4. Proof of Theorem 4

Proof. 

Suppose that $\mathcal{F}$ is not normal at $z_0$. According to Lemma 4 and Lemma 5, in order to prove the normality of the family of functions $\mathcal{F}$, we only need to prove that $\mathcal{F}$ is normal in the region D when $H\left(z\right)$ has at least one multiple zero and $P\left(z_0\right)=0$. When $P\left(z_0\right)=0$, there must exist a sufficiently small positive number $\delta$ such that for any $z\in D_{\delta }=\{z:|z-z_0|<\delta \}$, the function $P\left(z\right)$ has a unique zero at $z_0$ and no poles within the neighborhood of $z_0$. To simplify subsequent discussions without sacrificing generality, we set $z_0=0$. At this time, $P\left(z\right)$ can be expressed as $P\left(z\right)=z^{t}b\left(z\right)$. Here, t is a positive integer satisfying $1\le t\le m$, $b\left(z\right)$ satisfies $b\left(z_0\right)=b\left(0\right)=1$, and $b\left(z\right)\ne 0$ in the region $D_{\delta }$ (otherwise, we can just construct a new family of functions $G=\left\{g\right|g=f(z+z_0),f\in \mathcal{F},z\in D_{\delta }\}$.) Based on the above settings, we will further discuss this situation in detail using two cases.

Case 1: $H\left(z\right)$ has a multiple zero at $z=0$.

Let

$$H\left(z\right)=a_p{z}^p+\cdots +a_q{z}^q,$$

(10)

where $p,q\ge 2$ and $a_q\ne 0$. Based on the above settings, according to Lemma 1, we have

$${\psi }_n\left(\zeta \right)=\frac{f_n(z_n+{\rho }_n\zeta )}{{\rho }_n^{\frac{t+k}{q+1}}}\stackrel{\chi }{\Rightarrow }\psi \left(\zeta \right).$$

(11)

Here, the function sequence $f_n$, the point sequence $z_n$, and the positive sequence ${\rho }_n$ have the corresponding properties in Lemma 1. On each compact subset of the complex plane $\mathbb{C}$, Expression (11) holds uniformly with respect to the spherical distance. Here, $\psi \left(\zeta \right)$ is a non-constant meromorphic function in $\mathbb{C}$ with zeros and poles that have multiplicities of at least $k+m$ and with a spherical derivative that is bounded and of order at most 2. We can consider two further cases.

Case 1.1: There exist subsequences of $z_n$ and ${\rho }_n$ (still denoted as $z_n$ and ${\rho }_n$ for simplicity) such that $\frac{z_n}{{\rho }_n}\to \infty$. Let

$${\mathrm{\Psi }}_n\left(\zeta \right)=\frac{f_n(z_n+z_n\zeta )}{z_n^{\frac{t+k}{q+1}}}.$$

(12)

Per (12), we have

$$\begin{array}{cc}\hfill z_n^{\frac{s(t+k)}{q+1}}{\mathrm{\Psi }}_n^s\left(\zeta \right)& =f_n^s(z_n+z_n\zeta ),(q\le s\le n),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill z_n^{\frac{(t+k)}{q+1}-i}{\mathrm{\Psi }}_n^{\left(i\right)}\left(\zeta \right)& =f_n^{\left(i\right)}(z_n+z_n\zeta ),(i=1,2,\cdots ,k).\hfill \end{array}$$

(14)

From (10), (12)–(14), we can get

$$\begin{array}{cc}\hfill & \frac{1}{z_n^t}[H\left(f_n(z_n+z_n\zeta )\right)f_n^{\left(k\right)}(z_n+z_n\zeta )-P(z_n+z_n\zeta )]\hfill \\ \hfill & =\frac{1}{z_n^t}[a_p{f}_n^p(z_n+z_n\zeta )+\cdots +a_q{f}_n^q(z_n+z_n\zeta )f_n^{\left(k\right)}(z_n+z_n\zeta )-{(z_n+z_n\zeta )}^{t}b(z_n+z_n\zeta )]\hfill \\ \hfill & =\frac{1}{z_n^t}[a_p{z}_n^{\frac{p(t+k)}{q+1}}{\mathrm{\Psi }}_n^p\left(\zeta \right)+\cdots +a_q{z}_n^{\frac{q(t+k)}{q+1}}{\mathrm{\Psi }}_n^q\left(\zeta \right)z_n^{\frac{(t+k)}{q+1}-k}{\mathrm{\Psi }}_n^{\left(k\right)}\left(\zeta \right)-{(z_n+z_n\zeta )}^{t}b(z_n+z_n\zeta )]\hfill \\ \hfill & =\sum _{s=q}^p{a}_s{z}_n^{\frac{(s-q)(t+k)}{q+1}}{\mathrm{\Psi }}_n^s\left(\zeta \right){\mathrm{\Psi }}_n^{\left(k\right)}\left(\zeta \right)-{(1+\zeta )}^{t}b(z_n+z_n\zeta ).\hfill \end{array}$$

(15)

Next, analogous to the proof approach of Lemma 4, we can demonstrate that the expression (15) possesses at most one zero in the complex plane. It is evident that every function within the sequence ${\mathrm{\Psi }}_n\left(\zeta \right)$ contains zeros and poles of multiplicity not lower than $k+m$. Initially, we shall establish the normality of the function family $\left\{{\mathrm{\Psi }}_n\left(\zeta \right)\right\}$ in the domain $\Delta =\{\zeta :|\zeta |<1\}$. Suppose that $\left\{{\mathrm{\Psi }}_n\left(\zeta \right)\right\}$ is not normal at a point ${\zeta }_0\in \Delta$. According to Lemma 1, there exists a sequence of points ${\zeta }_n\in \Delta$ satisfying ${\zeta }_n\to {\zeta }_0$, a subsequence selected from the original function sequence ${\mathrm{\Psi }}_n\left(\zeta \right)$ (for simplicity, still denoted as ${\mathrm{\Psi }}_n\left(\zeta \right)$) with each ${\mathrm{\Psi }}_n\left(\zeta \right)\in \Delta$, and a sequence of positive real numbers ${\eta }_n\to 0$ (as $n\to \infty$), such that the following holds:

$$G_n\left(\xi \right)=\frac{{\mathrm{\Psi }}_n({\zeta }_n+{\eta }_n\xi )}{{\eta }_n^{\frac{k}{q+1}}}\stackrel{\chi }{\Rightarrow }G\left(\xi \right).$$

(16)

On each compact subset of the complex plane $\mathbb{C}$, Expression (16) holds uniformly with respect to the spherical distance. Here, $G\left(\xi \right)$ is a non-constant meromorphic function in $\mathbb{C}$ with zeros and poles that have multiplicities of at least $k+m$ and with a spherical derivative that is bounded and of order at most 2. On an arbitrary compact subset of $\mathbb{C}$ that excludes the poles of $G\left(\xi \right)$, by virtue of (16) it follows that

$$\begin{array}{cc}\hfill {\eta }_n^{\frac{ks}{q+1}}{G}_n^s\left(\xi \right)& ={\mathrm{\Psi }}_n^s({\zeta }_n+{\eta }_n\xi ),(q\le s\le k),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\eta }_n^{\frac{k}{q+1}-i}{G}_n^{\left(i\right)}\left(\xi \right)& ={\mathrm{\Psi }}_n^{\left(i\right)}({\zeta }_n+{\eta }_n\xi ),(i=1,2,\cdots ,k).\hfill \end{array}$$

(18)

Using (16)–(18), we get

$$\begin{array}{cc}\hfill & \sum _{s=q}^p{a}_s{z}_n^{\frac{(s-q)(t+k)}{q+1}}{\mathrm{\Psi }}_n^s({\zeta }_n+{\eta }_n\xi ){\mathrm{\Psi }}_n^{\left(k\right)}({\zeta }_n+{\eta }_n\xi )-{(1+({\zeta }_n+{\eta }_n\xi ))}^{t}b(z_n+z_n({\zeta }_n+{\eta }_n\xi ))\hfill \\ \hfill & =\sum _{s=q}^p{a}_s{z}_n^{\frac{(s-q)(t+k)}{q+1}}{\eta }_n^{\frac{k(s-q)}{q+1}}{G}_n^s\left(\xi \right)G_n^{\left(k\right)}\left(\xi \right)-{(1+({\zeta }_n+{\eta }_n\xi ))}^{t}b(z_n+z_n({\zeta }_n+{\eta }_n\xi ))\hfill \\ \hfill & \to a_q{G}^q\left(\xi \right)G^{\left(k\right)}\left(\xi \right)-{(1+{\zeta }_0)}^t.\hfill \end{array}$$

(19)

Because ${\zeta }_0\in \Delta$, we have $(1+{\zeta }_0)\ne 0$. Similar to the discussion of Lemmas 4 and 5 earlier, we are able to demonstrate that Expression (19) is incapable of being identically zero, and additionally that the expression possesses at most a single zero. On the contrary, in accordance with Lemma 2, it is revealed that Expression (19) contains no less than two distinct zeros, an outcome that flatly contradicts the ongoing derivation. It follows that the family of functions $\left\{{\mathrm{\Psi }}_n\left(\zeta \right)\right\}$ is normal in the region $\Delta$. Therefore, there must exist a subsequence of $\left\{{\mathrm{\Psi }}_n\left(\zeta \right)\right\}$ (for the sake of convenience, still denoted as $\left\{{\mathrm{\Psi }}_n\left(\zeta \right)\right\}$) and a meromorphic function $\mathrm{\Psi }\left(\zeta \right)$ such that ${\mathrm{\Psi }}_n\left(\zeta \right)$ converges to $\mathrm{\Psi }\left(\zeta \right)$.

If $\mathrm{\Psi }\left(0\right)\ne \infty$, combining (11) and (12) on any compact subset of the complex plane $\mathbb{C}$ excluding the poles of $\psi \left(\zeta \right)$ and considering the condition $m+k-1-\frac{t+k}{q+1}>0$, we can deduce that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{\psi }^{(k+m-1)}\left(\zeta \right)& =\underset{n\to \infty }{lim}{\psi }_n^{(k+m-1)}\left(\zeta \right)=\underset{n\to \infty }{lim}\frac{f_n^{(k+m-1)}(z_n+{\rho }_n\zeta )}{{\rho }_n^{\frac{t+k}{q+1}-k-m+1}}\hfill \\ & =\underset{n\to \infty }{lim}{\left(\frac{{\rho }_n}{z_n}\right)}^{k+m-\frac{t+k}{q+1}-1}{\mathrm{\Psi }}_n^{(k+m-1)}\left(\frac{{\rho }_n}{z_n}\zeta \right)=0.\hfill \end{array}$$

It can be inferred that ${\psi }^{(k+m-1)}\left(\zeta \right)$ is identically zero, which indicates that $\psi \left(\zeta \right)$ is a polynomial that degree not exceeding $k+m-1$, which contradicts the fact that the multiplicity of zeros of $\psi \left(\zeta \right)$ is at least $k+m$.

If $\mathrm{\Psi }\left(0\right)=\infty$, then from

$$\frac{1}{{\mathrm{\Psi }}_n\left(\frac{{\rho }_n}{z_n}\zeta \right)}=\frac{z_n^{\frac{t+k}{q+1}}}{f_n(z_n+{\rho }_n\zeta )}\to \frac{1}{\mathrm{\Psi }\left(0\right)}=0$$

on any compact subset of the complex plane $\mathbb{C}$ excluding the poles of $\psi \left(\zeta \right)$, we get

$$\begin{array}{cc}\hfill & \frac{1}{\psi \left(\zeta \right)}=\underset{n\to \infty }{lim}\frac{1}{{\psi }_n\left(\zeta \right)}=\underset{n\to \infty }{lim}\frac{{\rho }_n^{\frac{t+k}{q+1}}}{f_n(z_n+{\rho }_n\zeta )}=\underset{n\to \infty }{lim}{\left(\frac{{\rho }_n}{z_n}\right)}^{\frac{t+k}{q+1}}\frac{1}{{\mathrm{\Psi }}_n\left(\frac{{\rho }_n}{z_n}\zeta \right)}=0.\hfill \end{array}$$

Therefore, $\psi \left(\zeta \right)$ is constantly ∞, which contradicts the preconditions for $\psi \left(\zeta \right)$.

Case 1.2: Here exist subsequences of $z_n$ and ${\rho }_n$ (still denoted as $z_n$ and ${\rho }_n$ for simplicity) such that $\frac{z_n}{{\rho }_n}\to \alpha$($\alpha \ne \infty$). Following (11), on any compact subset of the complex plane $\mathbb{C}$ excluding the poles of $\psi \left(\zeta \right)$, we have

$$\begin{array}{c}\hfill {\rho }_n^{\frac{s(t+k)}{q+1}}{\psi }_n^s\left(\zeta \right)=f_n^s(z_n+{\rho }_n\zeta ),(2\le q\le s\le n),\end{array}$$

(20)

$$\begin{array}{c}\hfill {\rho }_n^{\frac{(t+k)}{q+1}-i}{\psi }_n^{\left(i\right)}\left(\zeta \right)=f_n^{\left(i\right)}(z_n+{\rho }_n\zeta ),(i=1,2,\cdots ,k).\end{array}$$

(21)

From (10), (11), (20), and (21), we can obtain

$$\begin{array}{cc}\hfill & \frac{1}{{\rho }_n^t}\left[H\left(f_n(z_n+{\rho }_n\zeta )\right)f_n^{\left(k\right)}(z_n+{\rho }_n\zeta )-P(z_n+{\rho }_n\zeta )\right]\hfill \\ \hfill & =\frac{1}{{\rho }_n^t}\left[(a_p{f}_n^p(z_n+{\rho }_n\zeta )+\cdots +a_q{f}_n^q(z_n+{\rho }_n\zeta ))f_n^{\left(k\right)}(z_n+{\rho }_n\zeta )-{(z_n+{\rho }_n\zeta )}^{t}b(z_n+{\rho }_n\zeta )\right]\hfill \\ \hfill & =\sum _{s=q}^p{a}_s{\rho }_n^{\frac{(s-q)(t+k)}{q+1}}{\psi }_n^s\left(\zeta \right){\psi }_n^{\left(k\right)}\left(\zeta \right)-{(\frac{z_n}{{\rho }_n}+\zeta )}^{t}b(z_n+{\rho }_n\zeta )\hfill \\ \hfill & \to a_q{\psi }^q\left(\zeta \right){\psi }^{\left(k\right)}\left(\zeta \right)-{(\alpha +\zeta )}^t.\hfill \end{array}$$

(22)

We can find that Expression (22) does not vanish identically, and further that Expression (22) possesses at most a single zero in $\mathbb{C}$. Conversely, in light of Lemma 3, Expression (22) exhibits no fewer than two distinct zeros, an inconsistency that directly engenders a contradiction.

Case 2: The zero of $H\left(z\right)$ is at $z=b$ (where b is a finite non-zero complex number).

We can set

$$\begin{array}{c}\hfill H\left(z\right)=a_p{(z-b)}^p+\cdots +a_q{(z-b)}^q=Q_1\left(z\right){(z-b)}^q,(2\le q\le p),\end{array}$$

(23)

among which $p,q\ge 2$ are positive integers, $a_q$ and b are both finite non-zero complex numbers, and $Q_1\left(z\right)$ is a non-zero polynomial. According to Lemma 1, we have

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_n\left(\zeta \right)=f_n(z_n+{\rho }_n\zeta )\stackrel{\chi }{\Rightarrow }\mathrm{\Gamma }\left(\zeta \right).\end{array}$$

(24)

On each compact subset of the complex plane $\mathbb{C}$, Expression 11 holds uniformly with respect to the spherical distance. Here, $\mathrm{\Gamma }\left(\zeta \right)$ is a non-constant meromorphic function in $\mathbb{C}$ with zeros and poles that have multiplicities of at least $k+m$ and with a spherical derivative that is bounded and of order at most 2. On an arbitrary compact subset of $\mathbb{C}$ that excludes the poles of $\mathrm{\Gamma }\left(\zeta \right)$, by virtue of (24) it follows that

$$\begin{array}{c}\hfill {\rho }_n^{-i}{\mathrm{\Gamma }}_n^{\left(i\right)}\left(\zeta \right)=f_n^{\left(i\right)}(z_n+{\rho }_n\zeta ).\end{array}$$

(25)

By (23)–(25), we get

$$\begin{array}{cc}\hfill & {\rho }_n^k[P\left(f_n(z_n+{\rho }_n\zeta )\right)f_n^{\left(k\right)}(z_n+{\rho }_n\zeta )-P(z_n+{\rho }_n\zeta )]\hfill \\ \hfill & =H\left({\mathrm{\Gamma }}_n\left(\zeta \right)\right){\mathrm{\Gamma }}_n^{\left(k\right)}\left(\zeta \right)-{\rho }_n^{k}P(z_n+{\rho }_n\zeta )\hfill \\ \hfill & \to P\left(\mathrm{\Gamma }\left(\zeta \right)\right){\mathrm{\Gamma }}^{\left(k\right)}\left(\zeta \right)=Q_1\left(\mathrm{\Gamma }\left(\zeta \right)\right){(\mathrm{\Gamma }\left(\zeta \right)-b)}^q{\mathrm{\Gamma }}^{\left(k\right)}\left(\zeta \right).\hfill \end{array}$$

(26)

Similar to the derivation of Lemma 4, we first establish that the function (26) is not identically zero and has at most one zero. This further implies that the number of zeros of $\left(\mathrm{\Gamma }\right(\zeta )-b)\mathrm{\Gamma }\left(\zeta \right)$ is also at most one. This means that among the values 0 and b, at most one of them is attained by the function $\mathrm{\Gamma }\left(\zeta \right)$. To simplify the derivation, we suppose without loss of generality that $\mathrm{\Gamma }\left(\zeta \right)$ never assumes the value of zero. By invoking this supposition and applying the First and Second Fundamental Theorems of Nevanlinna, the subsequent conclusions can be deduced:

$$\begin{array}{cc}\hfill T(r,\mathrm{\Gamma })& <\overline{N}(r,\mathrm{\Gamma })+\overline{N}\left(r,\frac{1}{\mathrm{\Gamma }}\right)+\overline{N}\left(r,\frac{1}{\mathrm{\Gamma }-b}\right)+S(r,\mathrm{\Gamma })\hfill \\ \hfill & \le \overline{N}(r,\mathrm{\Gamma })+logr+S(r,\mathrm{\Gamma })\le 2logr+S(r,\mathrm{\Gamma }).\hfill \end{array}$$

This conclusion implies that $\mathrm{\Gamma }\left(\zeta \right)$ cannot be a transcendental function, and obviously cannot be a polynomial function. Therefore, $\mathrm{\Gamma }\left(\zeta \right)$ can only be a non-polynomial rational function. Further, combining the fact that the multiplicities of both zeros and poles of $\mathrm{\Gamma }\left(\zeta \right)$ are at least $k+m\ge 2$, we can assume its form to be $\mathrm{\Gamma }\left(\zeta \right)=\frac{A}{{(\zeta -\beta )}^2}$ (where $A\ne 0$). Through calculation, it is known that $\mathrm{\Gamma }\left(\zeta \right)-b=\frac{A}{{(\zeta -\beta )}^2}-b$ has exactly two zeros $\zeta =\beta \pm \sqrt{\frac{A}{b}}$, which clearly contradicts the premise that $\mathrm{\Gamma }\left(\zeta \right)-b$ has at most one zero.

Synthesizing the above derivation process, we can finally conclude that $\mathcal{F}$ is normal in the region D; the proof is completed.

5. Proof of Theorem 3

Proof. 

To prove the normality of $\mathcal{F}$ at an arbitrary point $z_0\in D$, consider an arbitrary function $f\in \mathcal{F}$, leading to two cases. First, if $H\left(f_n\left(z_0\right)\right)f_n^{\left(k\right)}\left(z_0\right)\ne P\left(z_0\right)$, then there exists ${\delta }_1>0$ such that $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)-P\left(z\right)$ has no zeros in $D_{{\delta }_1}=\{z:|z-z_0|<{\delta }_1\}$. Second, if $H\left(f_n\left(z_0\right)\right)f_n^{\left(k\right)}\left(z_0\right)=P\left(z_0\right)$, then there exists ${\delta }_2>0$ such that $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)-P\left(z\right)$ has only one zero at $z_0$ in $D_{{\delta }_2}=\{z:|z-z_0|<{\delta }_2\}$. Define $\delta =min\{{\delta }_1,{\delta }_2\}$; then, $H\left(f_n\left(z\right)\right)f_n^{\left(k\right)}\left(z\right)-P\left(z\right)$ has at most one zero in $D_{\delta }$. Per the sharing condition, this holds for any $g\in \mathcal{F}$. Applying Theorem 4, $\mathcal{F}$ is normal in $D_{\delta }$, and hence at $z_0$. Because $z_0$ is arbitrary in D, $\mathcal{F}$ is normal in D, proving Theorem 3.

6. Conclusions

The integration of normal families of meromorphic functions with shared values and shared functions constitutes one of the core research directions in the theory of normal families. Current investigations in this field primarily proceed along two avenues. The first involves the optimization and extension of the structure of differential expressions associated with meromorphic functions, while the second focuses on the adjustment and refinement of the types and constraints of shared objects. For instance, within the classical framework of shared value research, one can explore more general normality criteria by replacing simple higher-order derivative forms with differential polynomials or by extending shared constants to more general function forms such as polynomials or holomorphic functions.

Building upon existing research findings, this paper conducts systematic generalization and improvement along the aforementioned two research paths. Regarding the structure of differential expressions, we extend the first-order differential polynomial $H\left(f\right)f^{\prime }$ in Theorem 1 to an arbitrary k-th order differential polynomial $H\left(f\right)f^{\left(k\right)}$, thereby overcoming the limitation of applicability restricted to first-order derivatives. Simultaneously, we weaken the structural constraint on the polynomial H, relaxing the strict condition of $degH\ge 3$ or $degH=2$ with a single zero to the more flexible requirement of $degH\ge 2$, which significantly broadens the scope of applicable differential polynomials.

In terms of shared objects, we generalize the original shared constant b to an arbitrary non-identically zero holomorphic function $P\left(z\right)$ and incorporate the constraint that the multiplicity of its zeros is at most m. Correspondingly, we adjust the condition on the multiplicity of zeros and poles of functions in the family $\mathcal{F}$ to be at least $k+m$, further enhancing the rationality of the sharing mechanism.

Through the aforementioned improvements, this paper establishes two main results, namely Theorems 3 and 4. These theorems respectively present normality criteria for families of meromorphic functions based on the sharing of a holomorphic function $P\left(z\right)$ by k-th order differential polynomials and under the constraint of the number of zeros of the difference between the differential polynomial and $P\left(z\right)$. These findings significantly improve the theoretical universality and application value of the relevant conclusions.

7. Discussion

The proofs of Corollaries 1 and 2 in this paper can be completed using methods entirely analogous to those employed for Theorems 3 and 4; thus, the detailed proof processes for both corollaries are omitted herein.

Example 4 in this paper has verified the following: if the shared object (a holomorphic function $P\left(z\right)$) in the theorems is replaced with a general meromorphic function $h\left(z\right)$ (e.g., taking $h\left(z\right)=\frac{1}{z^{q+3}}$), then the original normality conclusion will fail. Further analysis indicates that the failure of this conclusion may stem from the absence of explicit constraints on the multiplicity of poles of $h\left(z\right)$, the polar behavior of which consequently destroys the balance of growth within the function family. Based on this observation, the following conjecture is proposed.

Let D be a domain, $\mathcal{F}$ a family of meromorphic functions in D, $k,m\ge 1$ positive integers, and $h\left(z\right)$ a non-identically zero meromorphic function in D satisfying the requirements that the multiplicity of its poles be at most a fixed positive integer t and the multiplicity of its zeros be at most m. If for any $f\in \mathcal{F}$, the multiplicity of zeros of f is at least $k+m+t$, the multiplicity of poles of f is at least $2+t$, and the differential polynomial $H\left(f\right)f^{\left(k\right)}-h\left(z\right)$ has at most one distinct zero in D, then $\mathcal{F}$ is normal in D.

Inspired by the classic result proposed by Schwick in 1992 (i.e., a family of meromorphic functions is normal if it shares three values), the research perspective can be further extended as follows: even without requiring the function family to share all given objects, the partial sharing of multiple holomorphic functions (i.e., a subset of the set of shared objects) may still impose sufficiently strong constraints on the value-taking and derivative behavior of the function family, thereby ensuring its normality. Accordingly, the following conjecture is put forward.

Let D be a domain, $\mathcal{F}$ a family of meromorphic functions in D, $k\ge 1$ a positive integer, and $P_1\left(z\right),P_2\left(z\right),P_3\left(z\right)$ three mutually distinct non-identically zero holomorphic functions in D. If for any $f,g\in \mathcal{F}$ the differential polynomials $H\left(f\right)f^{\left(k\right)}$ and $H\left(g\right)g^{\left(k\right)}$ share at least two elements of the set $\{P_1\left(z\right),P_2\left(z\right),P_3\left(z\right)\}$ in D and the multiplicity of zeros of each $f\in \mathcal{F}$ is at least $k+1$ while the multiplicity of its poles is at least 2, then $\mathcal{F}$ is normal in D.

It should be noted that Theorems 3 and 4 and Corollaries 1 and 2 all impose the constraint that the differential polynomial $H\left(f\right)f^{\left(k\right)}-P\left(z\right)$ have at most one distinct zero in D. This constraint has not yet covered the more general case where the polynomial has at most n distinct zeros (where $n\ge 2$ is a fixed integer). For instance, if the aforementioned constraint is relaxed to the differential polynomial $H\left(f\right)f^{\left(k\right)}-P\left(z\right)$ having at most two distinct zeros in D, no clear answer has been obtained thus far as to whether the normality conclusion for the function family still holds. Currently, we have not been able to determine whether there exists a critical threshold N such that the normality conclusion holds when the number of allowed distinct zeros $n\le N$ but fails when $n>N$. For this more generalized problem, we have not yet been able to provide a rigorous mathematical proof.