# A Normal Criterion Concerning Sequence of Functions and their Differential
Polynomials

Nikhil Bharti  Department of Mathematics
University of Jammu
Jammu-180006, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, a normality criterion concerning a sequence of meromorphic
functions and their differential polynomials is obtained. Precisely, we have
proved: Let $\left\\{f_{j}\right\\}$ be a sequence of meromorphic functions in
the open unit disk $\mathbb{D}$ such that, for each $j,$ $f_{j}$ has poles of
multiplicity at least $m,~{}m\in\mathbb{N}.$ Let $\left\\{h_{j}\right\\}$ be a
sequence of holomorphic functions in $\mathbb{D}$ such that $h_{j}\rightarrow
h$ locally uniformly in $\mathbb{D},$ where $h$ is holomorphic in $\mathbb{D}$
and $h\not\equiv 0.$ Let $Q[f_{j}]$ be a differential polynomial of $f_{j}$
having degree $\lambda_{Q}$ and weight $\mu_{Q}.$ If, for each $j,$
$f_{j}(z)\neq 0$ and $Q[f_{j}]-h_{j}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$
zeros, ignoring multiplicities, in $\mathbb{D},$ then $\left\\{f_{j}\right\\}$
is normal in $\mathbb{D}.$

###### Key words and phrases:

Meromorphic functions, Normal families, Differential polynomials, Value
distribution theory

###### 1991 Mathematics Subject Classification:

Primary 30D45; Secondary 30D30, 30D35, 30M04

##  1\. Introduction

In what follows, $\mathcal{H}(D)$ and $\mathcal{M}(D)$ are the classes of all
holomorphic and meromorphic functions in the domain $D\subseteq\mathbb{C},$
respectively. A family $\mathcal{F}\subset\mathcal{M}(D)$ is said to be normal
in $D$ if every sequence of functions in $\mathcal{F}$ has a subsequence which
converges locally uniformly in $D$ with respect to the spherical metric to a
limit function which is either meromorphic in $D$ or the constant $\infty.$ In
case, if $\mathcal{F}\subset\mathcal{H}(D),$ then the Euclidean metric can be
substituted for the spherical metric (see [32, 41]). The idea of normal family
is attributed to Paul Montel [23, 24, 25]. Ever since its creation, the theory
of normal families has been a cornerstone of complex analysis with far
reaching applications in dynamics of rational as well as transcendental maps,
function theory of one and several variables, bicomplex analysis, complex
projective geometry, functional analysis etc. (see [1, 2, 6, 7, 16, 21, 38,
41]). The theory of normal families continues to be an active area of
research.

The main purpose of this paper is to study the normality of a sequence of non-
vanishing meromorphic functions in a domain $D\subseteq\mathbb{C}$ whose
differential polynomials have non-exceptional holomorphic functions in $D.$
For $f,~{}g\in\mathcal{M}(D),$ if $f(z)-g(z)\neq 0$ in $D,$ then $g$ is said
to be an exceptional function of $f$ in $D.$ On the other hand, if there exist
at least one $z\in D$ for which $f(z)-g(z)=0,$ then $g$ is said to be a non-
exceptional function of $f$ in $D.$ If $g$ happens to be a constant, say $k,$
then $k$ is said to be an exceptional (respectively, non-exceptional) value of
$f$ in $D.$

######  Definition 1.1.

[17] Let $k\in\mathbb{N}$ and $n_{0},n_{1},\ldots,n_{k}$ be non-negative
integers, not all zero. Let $f\in\mathcal{M}(D).$ Then the product

$M[f]:=a\cdot\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^{n_{j}}$

is called a differential monomial of $f,$ where $a~{}(\not\equiv
0,\infty)\in\mathcal{M}(D).$ If $a\equiv 1,$ then $M[f]$ is said to be a
normalized differential monomial of $f.$ The quantities

$\lambda_{M}:=\sum\limits_{j=0}^{k}n_{j}\mbox{ and
}\mu_{M}:=\sum\limits_{j=0}^{k}(j+1)n_{j}$

are called the degree and weight of the differential monomial $M[f],$
respectively.

For $1\leq i\leq m,$ let
$M_{i}[f]=\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^{n_{ji}}$ be $m$
differential monomials of $f.$ Then the sum

$P[f]:=\sum\limits_{i=1}^{m}a_{i}M_{i}[f]$

is called a differential polynomial of $f$ and the quantities

$\lambda_{P}:=\max\left\\{\lambda_{M_{i}}:1\leq i\leq m\right\\}\mbox{ and
}\mu_{P}:=\max\left\\{\mu_{M_{i}}:1\leq i\leq m\right\\}$

are called the degree and weight of the differential polynomial $P[f],$
respectively. If $\lambda_{M_{1}}=\lambda_{M_{2}}=\cdots=\lambda_{M_{m}},$
then $P[f]$ is said to be a homogeneous differential polynomial.

In the present work, we are concerned with the homogeneous differential
polynomials of the form

$Q[f]:=f^{x_{0}}(f^{x_{1}})^{(y_{1})}(f^{x_{2}})^{(y_{2})}\cdots(f^{x_{k}})^{(y_{k})},$
(1.1)

where $x_{0},~{}x_{1},~{}\ldots,~{}x_{k},~{}y_{1},~{}y_{2},~{}\ldots,~{}y_{k}$
are non-negative integers such that $x_{i}\geq y_{i}$ for
$i=1,~{}2,~{}\ldots,~{}k.$

Set $x^{\prime}=\sum\limits_{i=1}^{k}x_{i}$ and
$y^{\prime}=\sum\limits_{i=1}^{k}y_{i}.$ Further, we assume that $x_{0}>0$ and
$y^{\prime}>0.$ Using the generalized Leibniz rule for derivatives, one can
easily verify that

$(f^{x_{i}})^{(y_{i})}=\sum\limits_{n_{1}+n_{2}+\cdots+n_{x_{i}}=y_{i}}\frac{y_{i}!}{n_{1}!n_{2}!\cdots
n_{x_{i}}!}f^{(n_{1})}f^{(n_{2})}\cdots f^{(n_{x_{i}})},$

where $n_{i}$’s are non-negative integers. Thus the degree of $Q[f],$
$\lambda_{Q}=x_{0}+x^{\prime}$ and the weight of $Q[f],$
$\mu_{Q}=x_{0}+x^{\prime}+y^{\prime}=\lambda_{Q}+y^{\prime}.$

##  2\. A Glimpse of the Past

In $1907,$ Montel [23] proved that a locally uniformly bounded sequence of
holomorphic functions in a domain has a locally uniformly convergent
subsequence. Montel called a family of holomorphic functions exhibiting such
behaviour, a normal family in his subsequent paper [24] in $1912.$ However,
the fundamental ideas related to the study of normal convergence were
developed long before Montel. In $1860$s, Weierstrass established that a
uniform limit of a sequence of holomorphic functions in a domain is also
holomorphic (see [31]). In $1894,$ Stieltjes [33] proved that a uniformly
bounded sequence of holomorphic functions in a domain converging uniformly in
some open subset of the domain, also converges uniformly on compacta. These
results were further refined by Osgood [26], Vitali [34, 35] and Porter [29,
30] in early $1900$s. Montel’s definition of normality of a family of
holomorphic functions was later extended by Carath$\acute{e}$odary and Landau
[3] to a family of meromorphic functions by means of local uniform convergence
in the spherical metric. As a matter of fact, it was pointed out by Ostrowski
[27] that normality can be equivalently defined by using the spherical metric.
In $1967,$ Wu [38] studied normal families of holomorphic mappings between
complex manifolds. Later, Fujimoto [16] introduced the concept of
meromorphically normal families of meromorphic mappings in complex projective
spaces. In $1969,$ Drasin [15] introduced the value distribution theory of
Nevanlinna in the study of normal families. For a detailed account on the
history of the theory of normal families, the reader is referred to Zalcman
[41] and Beardon and Minda [2].

A major breakthrough in the theory of normal families happened in $1975$ when
Zalcman [40] proved his famous rescaling lemma, known as the Zalcman’s lemma,
which unlike normal families, gave a characterization of non-normal families
in a domain. This lemma provided an easy approach to obtain normality of a
family of functions using reductio ad absurdum and has revolutionized the
theory of normal families. Several extensions of Zalcman’s lemma in one and
several variables have been obtained (see Pang and Zalcman [28], Aladro and
Krantz [1], Charak and Sharma [8] and Dovbush [14] ). Kim and Krantz [21]
considered normal families of holomorphic functions and mappings on an
infinite dimensional Banach space. Charak et al. [6, 7] introduced normal
families of bicomplex holomorphic and meromorphic functions. Recently, Krantz
[22] investigated normal families of holomorphic mappings from the point of
view of invariant geometry and Chang et al. [5] studied normal families of
meromorphic functions in several complex variables.

##  3\. Motivation and Main Result

In [20, Problem 5.11], Hayman posed the following problem:

######  Problem 3.1.

Let $\mathcal{F}\subset\mathcal{M}(D)$ and $k$ be a positive integer. Suppose
that for each $f\in\mathcal{F},$ $f(z)\neq 0,$ $f^{(k)}\neq 1.$ Then what can
be said about the normality of $\mathcal{F}$ in $D?$

Gu [18] considered Problem 3.1 and confirmed that the family $\mathcal{F}$ is
indeed normal in $D.$ Subsequently, Yang [36] proved that the exceptional
value $1$ of $f^{(k)}$ can be replaced by an exceptional holomorphic function.
Chang [4] considered the case when $f^{(k)}-1$ has limited number of zeros and
obtained the normality of $\mathcal{F}.$ Later, Deng et al. [12] established
that there is no loss of normality even when $f^{(k)}-h$ has zeros for some
$h\in\mathcal{H}(D)$ as long as the number of zeros are bounded by the
constant $k.$ Chen et al. [9] took a sequence of exceptional holomorphic
functions instead of single exceptional holomorphic function. Recently, Deng
et al. [13] proved the following theorem concerning a sequence of meromorphic
functions:

######  Theorem 3.2.

Let $\left\\{f_{j}\right\\}\subset\mathcal{M}(D)$ and
$\left\\{h_{j}\right\\}\subset\mathcal{H}(D)$ be sequences of functions in
$D.$ Assume that $h_{j}\rightarrow h$ locally uniformly in $D,$ where
$h\in\mathcal{H}(D)$ and $h\not\equiv 0.$ Let $k$ be a positive integer. If,
for each $j,$ $f_{j}(z)\neq 0$ and $f_{j}^{(k)}-h_{j}(z)$ has at most $k$
distinct zeros, ignoring multiplicities, in $D,$ then $\left\\{f_{j}\right\\}$
is normal in $D.$

A natural question that comes in mind is that whether Theorem 3.2 remains
valid if the $k$-th derivative $f_{j}^{(k)}$ is replaced by a differential
polynomial of $f_{j}.$ Finding an answer to this question is the main
objective of this paper. Borrowing the ideas from Deng et al. [13], we obtain
the following result which gives an affirmative answer to our problem. Since
normality is a local property, one can always restrict the domain to the open
unit disk $\mathbb{D}.$

######  Theorem 3.3.

Let $\left\\{f_{j}\right\\}\subset\mathcal{M}(\mathbb{D}),$ be a sequence such
that, for each $j,$ $f_{j}$ has poles of multiplicity at least
$m,~{}m\in\mathbb{N}.$ Let
$\left\\{h_{j}\right\\}\subset\mathcal{H}(\mathbb{D})$ be such that
$h_{j}\rightarrow h$ locally uniformly in $\mathbb{D},$ where
$h\in\mathcal{H}(\mathbb{D})$ and $h\not\equiv 0.$ Let $Q[f_{j}]$ be a
differential polynomial of $f_{j}$ as defined in (1.1) having degree
$\lambda_{Q}$ and weight $\mu_{Q}.$ If, for each $j,$ $f_{j}(z)\neq 0$ and
$Q[f_{j}]-h_{j}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ zeros, ignoring
multiplicities, in $\mathbb{D},$ then $\left\\{f_{j}\right\\}$ is normal in
$\mathbb{D}.$

It is important to note that the multiplicity of poles of $f_{j}$ in Theorem
3.3 is considered only to relax the upper bound for the zeros of
$Q[f_{j}]-h_{j}$ in $\mathbb{D}.$ In fact, Theorem 3.3 can be proved without
taking the multiplicity of poles of $f_{j}$ into account. In such case, the
hypothesis that $Q[f_{j}]-h_{j}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$
zeros, ignoring multiplicities, in $\mathbb{D},$ is replaced by the hypothesis
that $Q[f_{j}]-h_{j}$ has at most $\mu_{Q}-1$ zeros, ignoring multiplicities,
in $\mathbb{D}.$

As a direct consequence of Theorem 3.3, we have

######  Corollary 3.4.

Let $\left\\{f_{j}\right\\}\subset\mathcal{M}(\mathbb{D})$ and
$\left\\{h_{j}\right\\}\subset\mathcal{H}(\mathbb{D})$ be such that
$h_{j}\rightarrow h$ locally uniformly in $\mathbb{D},$ where
$h\in\mathcal{H}(D)$ and $h\not\equiv 0.$ If, for each $j,$ $f_{j}(z)\neq 0$
and $Q[f_{j}](z)\neq h_{j}(z),$ then $\left\\{f_{j}\right\\}$ is normal in
$\mathbb{D}.$

In the following, we show that the condition ‘$f_{j}(z)\neq 0$’ in Theorem 3.3
is essential.

######  Example 3.5.

Consider a sequence $\left\\{f_{j}\right\\}\subset\mathcal{M}(\mathbb{D})$
given by $f_{j}(z)=jz,~{}j\in\mathbb{N},~{}j\geq 2.$ Let
$Q[f_{j}]:=f_{j}f_{j}^{\prime}$ so that $\mu_{Q}=3,$ and let $h_{j}(z)=z.$
Then $h_{j}\rightarrow z\not\equiv 0$ and $Q[f_{j}](z)-h_{j}(z)$ has at most
one zero in $\mathbb{D}.$ However, $\left\\{f_{j}\right\\}$ is normal in
$\mathbb{D}.$

Taking $h_{j}(z)=1/z$ in Example 3.7, we find that $h_{j}$ cannot be
meromorphic in $\mathbb{D}.$ Furthermore, the condition ‘$h\not\equiv 0$’ in
Theorem 3.3 cannot be dropped as demonstrated by the following example:

######  Example 3.6.

Let $\left\\{f_{j}\right\\}\subset\mathcal{M}(\mathbb{D})$ such that
$f_{j}(z)=e^{jz},~{}j\in\mathbb{N}$ and let $h_{j}\equiv 0$ so that
$h_{j}\rightarrow h\equiv 0.$ Let $Q[f_{j}]$ be any differential polynomial of
$f_{j}$ of the form (1.1). Clearly, $Q[f_{j}](z)-h(z)$ has no zero in
$\mathbb{D}.$ But the sequence $\left\\{f_{j}\right\\}$ is not normal in
$\mathbb{D}.$

The following example establishes the sharpness of the condition
‘$Q[f_{j}]-h_{j}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros in
$\mathbb{D}$’ in Theorem 3.3:

######  Example 3.7.

Let $\left\\{f_{j}\right\\}\subset\mathcal{M}(\mathbb{D})$ such that

$f_{j}(z)=\frac{1}{jz},~{}j\geq 3,~{}j\in\mathbb{N}$

and let $Q[f_{j}]:=f_{j}f_{j}^{\prime}.$ Then $\lambda_{Q}=2,$
$\mu_{Q}=3,~{}m=1$ and $Q[f_{j}](z)=-1/j^{2}z^{3}.$ Consider
$h_{j}(z)=1/(z-1)^{3}$ so that
$\left\\{h_{j}\right\\}\in\mathcal{H}(\mathbb{D})$ and $h_{j}\rightarrow
1/(z-1)^{3}\not\equiv 0.$ Then by simple calculations, one can easily see that
$Q[f_{j}](z)-h_{j}(z)$ has exactly $\mu_{Q}+\lambda_{Q}(m-1)=3$ distinct zeros
in $\mathbb{D}.$ However, the sequence $\left\\{f_{j}\right\\}$ is not normal
in $\mathbb{D}.$

##  4\. Auxiliary Results

What follow are the preparations for the proof of the main result. Here, we
assume that the reader is familiar with standard notations of Nevanlinna’s
value distribution theory like $m(r,f),~{}N(r,f),~{}T(r,f),~{}S(r,f)$ (see
[19, 37]). Recall that a function $g\in\mathcal{M}(\mathbb{C})$ is said to be
a small function of $f\in\mathcal{M}(\mathbb{C})$ if $T(r,g)=S(r,f)$ as
$r\rightarrow\infty,$ possibly outside a set of finite Lebesgue measure.

Notation: By $D_{r}(a),$ we mean an open disk in $\mathbb{C}$ with center $a$
and radius $r.$ $\mathbb{D}=D_{1}(0)$ is the open unit disk in $\mathbb{C}.$

The following lemma is an extension of the Zalcman-Pang Lemma due to Chen and
Gu [10] (cf. [28, Lemma 2]).

######  Lemma 4.1 (Zalcman-Pang Lemma).

Let $\mathcal{F}\subset\mathcal{M}(\mathbb{D})$ be such that each
$f\in\mathcal{F},$ has zeros of multiplicity at least $m$ and poles of
multiplicity at least $p.$ Let $-p<\alpha<m.$ If $\mathcal{F}$ is not normal
at $z_{0}\in\mathbb{D},$ then there exist sequences
$\left\\{f_{j}\right\\}\subset\mathcal{F},$
$\left\\{z_{j}\right\\}\subset\mathbb{D}$ satisfying $z_{j}\rightarrow z_{0}$
and positive numbers $\rho_{j}$ with $\rho_{j}\rightarrow 0$ such that the
sequence $\left\\{g_{j}\right\\}$ defined by

$g_{j}(\zeta)=\rho_{j}^{-\alpha}f_{j}(z_{j}+\rho_{j}\zeta)\rightarrow
g(\zeta)$

locally uniformly in $\mathbb{C}$ with respect to the spherical metric, where
$g$ is a non-constant meromorphic function on $\mathbb{C}$ such that for every
$\zeta\in\mathbb{C},$ $g^{\\#}(\zeta)\leq g^{\\#}(0)=1.$

We remark that if $f(z)\neq 0$ in $D$ for every $f\in\mathcal{F},$ then
$\alpha\in(-p,~{}+\infty).$ Likewise, if each $f\in\mathcal{F}$ does not have
any pole in $D,$ then $\alpha\in(-\infty,~{}m)$ and if $f(z)\neq 0,~{}\infty$
in $D,$ for every $f\in\mathcal{F},$ then $\alpha\in(-\infty,~{}+\infty).$

######  Lemma 4.2.

[11, Lemma 3] Let $\mathcal{F}\subset\mathcal{M}(\mathbb{D})$ and suppose that
$h\in\mathcal{H}(\mathbb{D})$ or $h\equiv\infty.$ Further, assume that, for
each $f\in\mathcal{F},$ $f(z)\neq h(z)$ in $\mathbb{D}.$ If $\mathcal{F}$ is
normal in $\mathbb{D}\setminus\left\\{0\right\\}$ but not normal in
$\mathbb{D},$ then there exists a sequence
$\left\\{f_{j}\right\\}\subset\mathcal{F}$ such that $f_{j}\rightarrow h$ in
$\mathbb{D}\setminus\left\\{0\right\\}.$

######  Proposition 4.3.

Let $f\in\mathcal{M}(\mathbb{C})$ be a transcendental function and let $Q[f]$
be a differential polynomial of $f$ as defined in (1.1) having degree
$\lambda_{Q}$ and weight $\mu_{Q}.$ Assume that $\psi~{}(\not\equiv
0,~{}\infty)$ is a small function of $f.$ Then

$\lambda_{Q}T(r,f)\leq\overline{N}(r,f)+(1+\mu_{Q}-\lambda_{Q})\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{Q[f]-\psi}\right)+S(r,f).$

###### Proof.

By definition of $Q[f],$ it is apparent that $Q[f]\not\equiv 0.$ Then from the
first fundamental theorem of Nevanlinna, we have

$\displaystyle\lambda_{Q}T(r,f)$
$\displaystyle=\lambda_{Q}~{}m\left(r,\frac{1}{f}\right)+\lambda_{Q}N\left(r,\frac{1}{f}\right)+O(1)$
$\displaystyle\leq
m\left(r,\frac{Q[f]}{f^{\lambda_{Q}}}\right)+m\left(r,\frac{1}{Q[f]}\right)+\lambda_{Q}N\left(r,\frac{1}{f}\right)+O(1).$
(4.1)

From the Nevanlinna’s theorem on logarithmic derivative, we find that

$m\left(r,\frac{Q[f]}{f^{\lambda_{Q}}}\right)=S(r,f).$

Thus from (4), we obtain

$\displaystyle\lambda_{Q}T(r,f)$ $\displaystyle\leq
m\left(r,\frac{1}{Q[f]}\right)+\lambda_{Q}N\left(r,\frac{1}{f}\right)+S(r,f)$
$\displaystyle=T(r,Q[f])-N\left(r,\frac{1}{Q[f]}\right)+\lambda_{Q}N\left(r,\frac{1}{f}\right)+S(r,f).$

Applying the second fundamental theorem of Nevanlinna for small functions to
$T(r,Q[f]),$ we get

$\displaystyle\lambda_{Q}T(r,f)$
$\displaystyle\leq\lambda_{Q}N\left(r,\frac{1}{f}\right)+\overline{N}(r,Q[f])+\overline{N}\left(r,\frac{1}{Q[f]}\right)+\overline{N}\left(r,\frac{1}{Q[f]-\psi}\right)$
$\displaystyle\qquad-N\left(r,\frac{1}{Q[f]}\right)+S(r,f)$
$\displaystyle=\lambda_{Q}N\left(r,\frac{1}{f}\right)+\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{Q[f]}\right)+\overline{N}\left(r,\frac{1}{Q[f]-\psi}\right)$
(4.2) $\displaystyle\qquad-N\left(r,\frac{1}{Q[f]}\right)+S(r,f).$

Since a zero of $f$ with multiplicity $m$ is also a zero of $Q[f]$ with
multiplicity at least $(m+1)\lambda_{Q}-\mu_{Q},$

$N\left(r,\frac{1}{Q[f]}\right)-\overline{N}\left(r,\frac{1}{Q[f]}\right)\geq\left[(m+1)\lambda_{Q}-\mu_{Q}-1\right]\overline{N}\left(r,\frac{1}{f}\right).$

Therefore, from (4), we obtain

$\displaystyle\lambda_{Q}T(r,f)$
$\displaystyle\leq\lambda_{Q}N\left(r,\frac{1}{f}\right)+\overline{N}(r,f)+\left[1+\mu_{Q}-(m+1)\lambda_{Q}\right]\overline{N}\left(r,\frac{1}{f}\right)$
$\displaystyle\qquad+\overline{N}\left(r,\frac{1}{Q[f]-\psi}\right)+S(r,f)$
$\displaystyle\leq\overline{N}(r,f)+(1+\mu_{Q}-\lambda_{Q})\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{Q[f]-\psi}\right)+S(r,f).$

∎

######  Corollary 4.4.

Let $f\in\mathcal{M}(\mathbb{C})$ be a transcendental function and let $Q[f]$
be a differential polynomial of $f$ as defined in (1.1). Assume that
$\psi~{}(\not\equiv 0,~{}\infty)$ is a small function of $f.$ If $f\neq 0,$
then $Q[f]-\psi$ has infinitely many zeros in $\mathbb{C}.$

###### Proof.

From Proposition 4.3, we have

$\displaystyle\lambda_{Q}T(r,f)$
$\displaystyle\leq\overline{N}(r,f)+(1+\mu_{Q}-\lambda_{Q})\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{Q[f]-\psi}\right)$
$\displaystyle\qquad+S(r,f).$ (4.3)

Since $f\neq 0,$ $\overline{N}(r,1/f)=0.$ Thus from (4), we obtain

$\displaystyle\lambda_{Q}T(r,f)$
$\displaystyle\leq\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{Q[f]-\psi}\right)+S(r,f).$

This implies that

$(\lambda_{Q}-1)T(r,F)\leq\overline{N}\left(r,\frac{1}{Q[F]-\psi}\right)+S(r,F).$

Since $\lambda_{Q}-1>0,$ it follows that $Q[F]-\psi$ has infinitely many zeros
in $\mathbb{C}.$ ∎

In [4], Chang proved that if $f$ is a non-constant rational function such that
$f\neq 0,$ then for $k\geq 1,$ $f^{(k)}-1$ has at least $k+1$ distinct zeros
in $\mathbb{C}.$ Using the method of Chang [4], Deng et al. [12] proved that
the constant $1$ can be replaced by a polynomial $p~{}(\not\equiv 0).$
Recently, Xie and Deng [39] sharpened the lower bound for the distinct zeros
of $f^{(k)}-p$ in $\mathbb{C}$ by introducing the multiplicity of poles of
$f.$ We pick up the arguments of Chang [4] and obtain the corresponding result
for differential polynomials as:

######  Proposition 4.5.

Let $f$ be a non-constant rational function, having poles of multiplicity at
least $m,$ $m\in\mathbb{N},$ and let $p~{}(\not\equiv 0)$ be a polynomial. Let
$Q[f]$ be a differential polynomial of $f$ as defined in (1.1) having degree
$\lambda_{Q}$ and weight $\mu_{Q}.$ Assume that $f\neq 0.$ Then $Q[f]-p$ has
at least $\mu_{Q}+\lambda_{Q}(m-1)$ distinct zeros in $\mathbb{C}.$

###### Proof.

Since $f\neq 0,$ it follows that $f$ cannot be a polynomial and so $f$ has at
least one pole. Therefore, we can write

$f(z)=\frac{C_{1}}{\prod\limits_{i=1}^{n}(z+\alpha_{i})^{n_{i}+m-1}}.$ (4.4)

Let

$p(z)=C_{2}\prod\limits_{i=1}^{l}(z+\beta_{i})^{l_{i}},$ (4.5)

where $C_{1},~{}C_{2}$ are non-zero constants; $l,~{}n,~{}n_{i}$ are positive
integers and $l_{i}$ are non-negative integers. Also, $\beta_{i}$ (when $1\leq
i\leq l$) are distinct complex numbers and $\alpha_{i}$ (when $1\leq i\leq n$)
are distinct complex numbers.

From (4.4), one can deduce that

$Q[f](z)=\frac{h_{Q}(z)}{\prod\limits_{i=1}^{n}(z+\alpha_{i})^{\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}}},$
(4.6)

where $h_{Q}$ is a polynomial of degree $(n-1)(\mu_{Q}-\lambda_{Q}).$

Also, it is easy to see that $Q[f]-p$ has at least one zero in $\mathbb{C}.$
Therefore, we can set

$Q[f](z)=p(z)+\frac{C_{3}\prod\limits_{i=1}^{q}(z+\gamma_{i})^{q_{i}}}{\prod_{i=1}^{n}(z+\alpha_{i})^{\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}}},$
(4.7)

where $C_{3}\in\mathbb{C}\setminus\left\\{0\right\\},$ $q_{i}$ are positive
integers and $\gamma_{i}$ ($1\leq i\leq q$) are distinct complex numbers.

Let $L=\sum\limits_{i=1}^{l}l_{i}$ and $N=\sum\limits_{i=1}^{n}n_{i}.$ Then
from (4.5), (4.6) and (4.7), we have

$C_{2}\prod\limits_{i=1}^{l}(z+\beta_{i})^{l_{i}}\prod_{i=1}^{n}(z+\alpha_{i})^{\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}}+C_{3}\prod\limits_{i=1}^{q}(z+\gamma_{i})^{q_{i}}=h_{Q}(z)$
(4.8)

From (4.8), we find that

$\displaystyle\sum\limits_{i=1}^{q}q_{i}$
$\displaystyle=\sum\limits_{i=1}^{n}\left[\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}\right]+\sum\limits_{i=1}^{l}l_{i}$
$\displaystyle=\lambda_{Q}N+n(m-1)\lambda_{Q}+n(\mu_{Q}-\lambda_{Q})+L$

and $C_{3}=-C_{2}.$

Also, from (4.8), we get

$\displaystyle\prod\limits_{i=1}^{l}(1+\beta_{i}r)^{l_{i}}\prod_{i=1}^{n}(1+\alpha_{i}r)^{\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}}-\prod\limits_{i=1}^{q}(1+\gamma_{i}r)^{q_{i}}$
$\displaystyle=r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L}b(r),$

where $b(r):=r^{(n-1)(\mu_{Q}-\lambda_{Q})}h_{Q}(1/r)/C_{2}$ is a polynomial
of degree is at most $(n-1)(\mu_{Q}-\lambda_{Q}).$ Furthermore, it follows
that

$\displaystyle\frac{\prod\limits_{i=1}^{l}(1+\beta_{i}r)^{l_{i}}\prod\limits_{i=1}^{n}(1+\alpha_{i}r)^{\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}}}{\prod\limits_{i=1}^{q}(1+\gamma_{i}r)^{q_{i}}}$
$\displaystyle=1+\frac{r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L}b(r)}{\prod\limits_{i=1}^{q}(1+\gamma_{i}r)^{q_{i}}}$
$\displaystyle=1+O\left(r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L}\right)\mbox{ as
}r\rightarrow 0.$ (4.9)

Taking logarithmic derivatives of both sides of (4), we obtain

$\displaystyle\sum\limits_{i=1}^{l}\frac{l_{i}\beta_{i}}{1+\beta_{i}r}$
$\displaystyle+\sum\limits_{i=1}^{n}\frac{\left[\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}\right]\alpha_{i}}{1+\alpha_{i}r}-\sum\limits_{i=1}^{q}\frac{q_{i}\gamma_{i}}{1+\gamma_{i}r}$
$\displaystyle=O\left(r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-1}\right)\mbox{ as
}r\rightarrow 0.$ (4.10)

Let
$S_{1}=\left\\{\beta_{1},\beta_{2},\ldots,\beta_{l}\right\\}\cap\left\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\right\\}$
and
$S_{2}=\left\\{\beta_{1},\beta_{2},\ldots,\beta_{l}\right\\}\cap\left\\{\gamma_{1},\gamma_{2},\ldots,\gamma_{q}\right\\}.$
Then we consider the following cases:
Case 1: $S_{1}=S_{2}=\emptyset.$
Let $\alpha_{n+i}=\beta_{i}$ when $1\leq i\leq l$ and

$N_{i}=\left\\{\begin{array}[]{cc}\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}&\mbox{if}~{}1\leq
i\leq n,\\\ l_{i-n}&\mbox{if}~{}n+1\leq i\leq n+l.\end{array}\right.$

Then (4) can be written as

$\sum\limits_{i=1}^{n+l}\frac{N_{i}\alpha_{i}}{1+\alpha_{i}r}-\sum\limits_{i=1}^{q}\frac{q_{i}\gamma_{i}}{1+\gamma_{i}r}=O\left(r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-1}\right)~{}\mbox{as
$r\rightarrow 0.$}$ (4.11)

Comparing the coefficients of
$r^{j},~{}j=0,1,\ldots,\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-2$ in (4.11), we find
that

$\sum\limits_{i=1}^{n+l}N_{i}\alpha_{i}^{j}-\sum\limits_{i=1}^{q}q_{i}\gamma_{i}^{j}=0,~{}\mbox{for
each}~{}j=1,2,\ldots,\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-1.$ (4.12)

Now, let $\alpha_{n+l+i}=\gamma_{i}$ for $1\leq i\leq q.$ Then from (4.12) and
the fact that $\sum\limits_{i=1}^{n+l}N_{i}-\sum\limits_{i=1}^{q}q_{i}=0,$ we
deduce that the system of equations

$\sum\limits_{i=1}^{n+l+q}\alpha_{i}^{j}x_{i}=0,~{}j=0,1,\ldots,\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-1,$
(4.13)

has a non-zero solution

$\left(x_{1},\ldots,x_{n+l},x_{n+l+1},\ldots,x_{n+l+q}\right)=\left(N_{1},\ldots,N_{n+l},-q_{1},\ldots,-q_{q}\right).$

This is possible only when the rank of the coefficient matrix of the system
(4.13) is strictly less than $n+l+q.$

Hence $\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L<n+l+q.$ Since
$N=\sum\limits_{i=1}^{n}n_{i}\geq n$ and $L=\sum\limits_{i=1}^{l}l_{i}\geq l,$
it follows that $q\geq\mu_{Q}+\lambda_{Q}(m-1).$
Case 2: $S_{1}\neq\phi$ and $S_{2}=\phi.$
We may assume, without loss of generality, that
$S_{1}=\left\\{\beta_{1},\beta_{2},\ldots,\beta_{s_{1}}\right\\}.$ Then
$\beta_{i}=\alpha_{i}$ for $1\leq i\leq s_{1}.$ Take $s_{3}=l-s_{1}.$
Subcase 2.1: $s_{3}\geq 1.$
Let $\alpha_{n+i}=\beta_{s_{1}+i}$ for $1\leq i\leq s_{3}$ and if $s_{1}<n,$
then let

$N_{i}=\left\\{\begin{array}[]{ccc}\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i}&\mbox{if
}1\leq i\leq s_{1},\\\ \lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}&\mbox{if
}s_{1}+1\leq i\leq n,\\\ l_{s_{1}-n+i}&\mbox{if }n+1\leq i\leq
n+s_{3}.\end{array}\right.$

If $s_{1}=n,$ then we take

$N_{i}=\left\\{\begin{array}[]{cc}\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i}&\mbox{if
}1\leq i\leq s_{1},\\\ l_{s_{1}-n+i}&\mbox{if }n+1\leq i\leq
n+s_{3}.\end{array}\right.$

Subcase 2.2: $s_{3}=0.$
If $s_{1}<n,$ then set

$N_{i}=\left\\{\begin{array}[]{cc}\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i}&\mbox{if
}1\leq i\leq s_{1},\\\ \lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}&\mbox{if
}s_{1}+1\leq i\leq n\end{array}\right.$

and if $s_{1}=n,$ then set
$N_{i}=\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i},~{}\mbox{for }1\leq
i\leq s_{1}=n.$

Thus (4) can be written as:

$\sum\limits_{i=1}^{n+s_{3}}\frac{N_{i}\alpha_{i}}{1+\alpha_{i}r}-\sum\limits_{i=1}^{q}\frac{q_{i}\gamma_{i}}{1+\gamma_{i}r}=O\left(r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-1}\right)~{}\mbox{as
$r\rightarrow 0,$}$

where $0\leq s_{3}\leq l-1.$ Proceeding in similar fashion as in Case 1, we
deduce that $q\geq\mu_{Q}+m-1.$
Case 3: $S_{1}=\phi$ and $S_{2}\neq\phi.$
We may assume, without loss of generality, that
$S_{2}=\left\\{\beta_{1},\beta_{2},\ldots,\beta_{s_{2}}\right\\}.$ Then
$\beta_{i}=\gamma_{i}$ for $1\leq i\leq s_{2}.$ Take $s_{4}=l-s_{2}.$
Subcase 3.1: $s_{4}\geq 1.$
Let $\gamma_{q+i}=\beta_{s_{2}+i}$ for $1\leq i\leq s_{4}$ and if $s_{2}<q,$
then set

$Q_{i}=\left\\{\begin{array}[]{ccc}q_{i}-l_{i}&\mbox{if }1\leq i\leq s_{2},\\\
q_{i}&\mbox{if }s_{2}+1\leq i\leq q,\\\ -l_{s_{2}-q+i}&\mbox{if }q+1\leq i\leq
q+s_{4}.\end{array}\right.$

If $s_{2}=q,$ then set

$Q_{i}=\left\\{\begin{array}[]{cc}q_{i}-l_{i}&\mbox{if }1\leq i\leq s_{2},\\\
-l_{s_{2}-q+i}&\mbox{if }q+1\leq i\leq q+s_{4}.\end{array}\right.$

Subcase 3.2: $s_{4}=0.$
If $s_{2}<q,$ then set

$Q_{i}=\left\\{\begin{array}[]{cc}q_{i}-l_{i}&\mbox{if }1\leq i\leq s_{2},\\\
q_{i}&\mbox{if }s_{2}+1\leq i\leq q\end{array}\right.$

and if $s_{2}=q,$ then set $Q_{i}=q_{i}-l_{i},~{}\mbox{for }1\leq i\leq
s_{2}=q.$

Thus (4) can be written as:

$\displaystyle\sum\limits_{i=1}^{n}\frac{\left[\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}\right]\alpha_{i}}{1+\alpha_{i}r}-\sum\limits_{i=1}^{q+s_{4}}\frac{Q_{i}\gamma_{i}}{1+\gamma_{i}r}$
$\displaystyle=O\left(r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-1}\right)\mbox{ as
}r\rightarrow 0,$

where $0\leq s_{4}\leq l-1.$ Proceeding in similar way as in Case 1, we deduce
that $q\geq\mu_{Q}+\lambda_{Q}(m-1).$
Case 4. $S_{1}\neq\phi$ and $S_{2}\neq\phi.$
We may assume, without loss of generality, that
$S_{1}=\left\\{\beta_{1},\beta_{2},\ldots,\beta_{s_{1}}\right\\},~{}S_{2}=\left\\{\gamma_{1},\gamma_{2},\ldots,\gamma_{s_{2}}\right\\}.$
Then $\beta_{i}=\alpha_{i}$ for $1\leq i\leq s_{1}$ and
$\gamma_{i}=\beta_{s_{1}+i}$ for $1\leq i\leq s_{2}.$ Take
$s_{5}=l-s_{2}-s_{1}.$
Subcase 4.1: $s_{5}\geq 1.$
Let $\alpha_{n+i}=u_{s_{1}+s_{2}+i}$ for $1\leq i\leq s_{5}$ and if $s_{1}<n,$
then set

$N_{i}=\left\\{\begin{array}[]{ccc}\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i}&\mbox{if
}1\leq i\leq s_{1},\\\ \lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}&\mbox{if
}s_{1}+1\leq i\leq n,\\\ l_{s_{1}+s_{2}-n+i}&\mbox{if }n+1\leq i\leq
n+s_{5}.\end{array}\right.$

If $s_{1}=n,$ then set

$N_{i}=\left\\{\begin{array}[]{cc}\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i}&\mbox{if
}1\leq i\leq s_{1},\\\ l_{s_{1}+s_{2}-n+i}&\mbox{if }n+1\leq i\leq
n+s_{5}.\end{array}\right.$

If $s_{2}<q,$ then set

$Q_{i}=\left\\{\begin{array}[]{cc}q_{i}-l_{s_{1}+i}&\mbox{if }1\leq i\leq
s_{2},\\\ q_{i}&\mbox{if }s_{2}+1\leq i\leq q\end{array}\right.$

and if $s_{2}=q,$ then set $Q_{i}=q_{i}-l_{s_{1}+i},~{}\mbox{for }1\leq i\leq
s_{2}.$
Subcase 4.2: $s_{5}=0.$
If $s_{1}<n,$ then set

$N_{i}=\left\\{\begin{array}[]{cc}\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i}&\mbox{if
}1\leq i\leq s_{1},\\\ \lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}&\mbox{if
}s_{1}+1\leq i\leq n.\end{array}\right.$

If $s_{1}=n,$ then set
$N_{i}=\lambda_{Q}(n_{i}+m-1)+\mu_{Q}-\lambda_{Q}+l_{i}~{}\mbox{for }1\leq
i\leq s_{1}.$

Also, if $s_{2}<q,$ then set

$Q_{i}=\left\\{\begin{array}[]{cc}q_{i}-l_{s_{1}+i}&\mbox{if }1\leq i\leq
s_{2},\\\ q_{i}&\mbox{if }s_{2}+1\leq i\leq q\end{array}\right.$

and if $s_{2}=q,$ then set $Q_{i}=q_{i}-l_{s_{1}+i},~{}\mbox{for }1\leq i\leq
s_{2}.$

Thus in both subcases, (4) can be written as

$\sum\limits_{i=1}^{n+s_{5}}\frac{N_{i}\alpha_{i}}{1+\alpha_{i}r}-\sum\limits_{i=1}^{q}\frac{Q_{i}\gamma_{i}}{1+\gamma_{i}r}=O\left(r^{\mu_{Q}+\lambda_{Q}(N+n(m-1)-1)+L-1}\right)~{}\mbox{as
$r\rightarrow 0,$}$

where $0\leq s_{5}\leq l-2.$ Proceeding in similar fashion as in Case 1, we
deduce that $q\geq\mu_{Q}+\lambda_{Q}(m-1).$ ∎

######  Lemma 4.6.

Let $\left\\{f_{j}\right\\}\subset\mathcal{M}(\mathbb{D})$ be a sequence of
non-vanishing functions, all of whose poles have multiplicities at least
$m,~{}m\in\mathbb{N}.$ Let
$\left\\{h_{j}\right\\}\subset\mathcal{H}(\mathbb{D})$ be such that
$h_{j}\rightarrow h$ locally uniformly in $\mathbb{D},$ where
$h\in\mathcal{H}(\mathbb{D})$ and $h(z)\neq 0$ in $\mathbb{D}.$ If, for each
$j,$ $Q[f_{j}]-h_{j}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ zeros, ignoring
multiplicities, in $\mathbb{D},$ then $\left\\{f_{j}\right\\}$ is normal in
$\mathbb{D}.$

###### Proof.

Without loss generality, suppose that $\left\\{f_{j}\right\\}$ is not normal
at $0\in\mathbb{D}.$ Then by Lemma 4.1, there exists a sequence of points
$\left\\{z_{j}\right\\}\subset\mathbb{D}$ with $z_{j}\rightarrow 0,$ a
sequence of positive real numbers satisfying $\rho_{j}\rightarrow 0,$ and a
subsequence of $\left\\{f_{j}\right\\},$ again denoted by
$\left\\{f_{j}\right\\}$ such that the sequence

$F_{j}(\zeta):=\frac{f_{j}(z_{j}+\rho_{j}\zeta)}{\rho_{j}^{(\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}\rightarrow
F(\zeta),$

spherically locally uniformly in $\mathbb{C},$ where
$F\in\mathcal{M}(\mathbb{C})$ is a non-constant and non-vanishing function
having poles of multiplicity at least $m.$ Clearly, $Q[F_{j}]\rightarrow Q[F]$
spherically uniformly in every compact subset of $\mathbb{C}$ disjoint from
poles of $F.$ Also, one can easily see that
$Q[F_{j}](\zeta)=Q[f_{j}](z_{j}+\rho_{j}\zeta).$ Thus, for every
$\zeta\in\mathbb{C}\setminus\left\\{F^{-1}(\infty)\right\\},$

$Q[f_{j}](z_{j}+\rho_{j}\zeta)-h_{j}(z_{j}+\rho_{j}\zeta)=Q[F_{j}](\zeta)-h_{j}(z_{j}+\rho_{j}\zeta)\rightarrow
Q[F](\zeta)-h(0)$

spherically locally uniformly. Since $F$ is non-constant and
$x_{0}>0,~{}x_{i}\geq y_{i}$ for all $i=1,~{}2,~{}\ldots,~{}k,$ by a result of
Grahl [17, Theorem 7], it follows that $Q[F]$ is non-constant. Next, we claim
that $Q[F]-h(0)$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ zeros in
$\mathbb{C}.$

Suppose, on the contrary, that $Q[F]-h(0)$ has $\mu_{Q}+\lambda_{Q}(m-1)$
distinct zeros in $\mathbb{C},$ say
$\zeta_{i},~{}i=1,~{}2,~{}\ldots,~{}\mu_{Q}+\lambda_{Q}(m-1).$ Then by
Hurwitz’s theorem, there exit sequences
$\zeta_{j,i},~{}i=1,~{}2,~{}\ldots,~{}\mu_{Q}+\lambda_{Q}(m-1)$ with
$\zeta_{j,i}\rightarrow\zeta_{i}$ such that for sufficiently large $j,$
$Q[f_{j}](z_{j}+\rho_{j}\zeta_{j,i})-h_{j}(z_{j}+\rho_{j}\zeta_{j,i})=0$ for
$i=1,~{}2,~{}\ldots,~{}\mu_{Q}+\lambda_{Q}(m-1).$ However, $Q[f_{j}]-h_{j}$
has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros in $\mathbb{D}.$ This
proves the claim. Now, from Corollary 4.4, it follows that $F$ must be a
rational function which contradicts Proposition 4.5. ∎

######  Proposition 4.7.

Let $t$ be a positive integer. Let
$\left\\{f_{j}\right\\}\subset\mathcal{M}(\mathbb{D})$ be a sequence of non-
vanishing functions, all of whose poles have multiplicities at least
$m,~{}m\in\mathbb{N}$ and let
$\left\\{h_{j}\right\\}\subset\mathcal{H}(\mathbb{D})$ be such that
$h_{j}\rightarrow h$ locally uniformly in $\mathbb{D},$ where
$h\in\mathcal{H}(\mathbb{D})$ and $h(z)\neq 0.$ If, for every j,
$Q[f_{j}](z)-z^{t}h_{j}(z)$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ zeros in
$\mathbb{D},$ then $\left\\{f_{j}\right\\}$ is normal in $\mathbb{D}.$

###### Proof.

In view of Lemma 4.6, it suffices to prove that $\mathcal{F}$ is normal at
$z=0.$ Since $h(z)\neq 0$ in $\mathbb{D},$ it can be assumed that $h(0)=1.$
Now, suppose, on the contrary, that $\left\\{f_{j}\right\\}$ is not normal at
$z=0.$ Then by Lemma 4.1, there exists a subsequence of
$\left\\{f_{j}\right\\},$ which, for simplicity, is again denoted by
$\left\\{f_{j}\right\\},$ a sequence of points
$\left\\{z_{j}\right\\}\subset\mathbb{D}$ with $z_{j}\rightarrow 0$ and a
sequence of positive real numbers satisfying $\rho_{j}\rightarrow 0$ such that
the sequence

$F_{j}(\zeta):=\frac{f_{j}(z_{j}+\rho_{j}\zeta)}{\rho_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}\rightarrow
F(\zeta)$

spherically locally uniformly in $\mathbb{C},$ where
$F\in\mathcal{M}(\mathbb{C})$ is a non-constant function. Also, since each
$f_{j}$ is non-vanishing, it follows that $F$ is non-vanishing. We now
distinguish two cases.
Case 1: Suppose that there exists a subsequence of $z_{j}/\rho_{j},$ again
denoted by $z_{j}/\rho_{j},$ such that $z_{j}/\rho_{j}\rightarrow\infty$ as
$j\rightarrow\infty.$

Define

$g_{j}(\zeta):=z_{j}^{-(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}f_{j}(z_{j}+z_{j}\zeta).$

Then an elementary computation shows that

$Q[g_{j}](\zeta)=z_{j}^{-t}Q[f_{j}](z_{j}+z_{j}\zeta),$

and hence

$\displaystyle Q[f_{j}](z_{j}+z_{j}\zeta)$
$\displaystyle-(z_{j}+z_{j}\zeta)^{t}h_{j}(z_{j}+z_{j}\zeta)$
$\displaystyle=z_{j}^{t}Q[g_{j}](\zeta)-(z_{j}+z_{j}\zeta)^{t}h_{j}(z_{j}+z_{j}\zeta)$
$\displaystyle=z_{j}^{t}\left[Q[g_{j}](\zeta)-(1+\zeta)^{t}h_{j}(z_{j}+z_{j}\zeta)\right].$

Since $(1+\zeta)^{t}h_{j}(z_{j}+z_{j}\zeta)\rightarrow(1+\zeta)^{t}\neq 0$ in
$\mathbb{D},$ and
$Q[f_{j}](z_{j}+z_{j}\zeta)-z_{j}^{t}(1+\zeta)^{t}h_{j}(z_{j}+z_{j}\zeta)$ has
at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ zeros in $\mathbb{D},$ by Lemma 4.6, it
follows that $\left\\{g_{j}\right\\}$ is normal in $\mathbb{D}$ and so there
exists a subsequence of $\left\\{g_{j}\right\\},$ again denoted by
$\left\\{g_{j}\right\\},$ such that $g_{j}\rightarrow g$ spherically locally
uniformly in $\mathbb{D},$ where $g\in\mathcal{M}(\mathbb{D})$ or
$g\equiv\infty.$ If $g\equiv\infty,$ then

$\displaystyle F_{j}(\zeta)$
$\displaystyle=\rho_{j}^{-(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}f_{j}(z_{j}+\rho_{j}\zeta)$
$\displaystyle=\left(\frac{z_{j}}{\rho_{j}}\right)^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}z_{j}^{-(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}f_{j}(z_{j}+\rho_{j}\zeta)$
$\displaystyle=\left(\frac{z_{j}}{\rho_{j}}\right)^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}g_{j}\left(\frac{\rho_{j}}{z_{j}}\zeta\right)$

converges spherically locally uniformly to $\infty$ in $\mathbb{C}$ showing
that $F\equiv\infty,$ a contradiction to the fact that $F$ is non-constant.
Since $g_{j}(\zeta)\neq 0,$ it follows that either $g(\zeta)\neq 0$ or
$g\equiv 0.$ If $g(\zeta)\neq 0,$ then by the previous argument, we find that
$F\equiv\infty,$ a contradiction. If $g\equiv 0,$ then choose $n\in\mathbb{N}$
such that $n+1>(t+\mu_{Q}-\lambda_{Q})/(\lambda_{Q}).$ Thus, for each
$\zeta\in\mathbb{C},$ we have

$\displaystyle F_{j}^{(n+1)}(\zeta)$
$\displaystyle=\rho_{j}^{-(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}+n+1}f_{j}^{(n+1)}(z_{j}+\rho_{j}\zeta)$
$\displaystyle=\left(\frac{\rho_{j}}{z_{j}}\right)^{-(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}+n+1}g_{j}^{(n+1)}\left(\frac{\rho_{j}}{z_{j}}\zeta\right).$

Therefore, $F_{j}^{(n+1)}(\zeta)\rightarrow 0$ spherically uniformly which
implies that $F$ is a polynomial of degree at most $n,$ a contradiction to the
fact that $F$ is non-constant and non-vanishing meromorphic function.
Case 2: Suppose that there exists a subsequence of $z_{j}/\rho_{j},$ again
denoted by $z_{j}/\rho_{j},$ such that $z_{j}/\rho_{j}\rightarrow\alpha$ as
$j\rightarrow\infty,$ where $\alpha\in\mathbb{C}.$ Then

$G_{j}(\zeta)=\rho_{j}^{-(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}f_{j}(\rho_{j}\zeta)=F_{j}\left(\zeta-\frac{z_{j}}{\rho_{j}}\right)\rightarrow
F(\zeta-\alpha):=G(\zeta),$

spherically locally uniformly in $\mathbb{C}.$ Clearly, $G(\zeta)\neq 0.$
Also, it is easy to see that
$Q[G_{j}](\zeta)=\rho_{j}^{-t}Q[f_{j}](\rho_{j}\zeta).$ Thus,

$Q[G_{j}](\zeta)-\zeta^{t}h_{j}(\rho_{j}\zeta)=\frac{Q[f_{j}](\rho_{j}\zeta)-(\rho_{j}\zeta)^{t}h_{j}(\rho_{j}\zeta)}{\rho_{j}^{t}}\rightarrow
Q[G](\zeta)-\zeta^{t}$

spherically uniformly in every compact subset of $\mathbb{C}$ disjoint from
the poles of $G.$ Clearly, $Q[G](\zeta)\not\equiv\zeta^{t},$ otherwise $G$ has
to be a polynomial, which is not possible since $G(\zeta)\neq 0.$ Since
$Q[f_{j}](\rho_{j}\zeta)-(\rho_{j}\zeta)^{t}h_{j}(\rho_{j}\zeta)$ has at most
$\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros in $\mathbb{D},$ it follows that
$Q[G](\zeta)-\zeta^{m}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct
zeros in $\mathbb{C}$ and hence by Corollary 4.4, $G$ must be a rational
function. However, this contradicts Proposition 4.5. Hence $\mathcal{F}$ is
normal in $\mathbb{D}.$ ∎

##  5\. Proof of Main Result

###### Proof of Theorem 3.3.

By virtue of Lemma 4.6, it is sufficient to prove the normality of
$\left\\{f_{j}\right\\}$ at points $z\in\mathbb{D}$ where $h(z)=0.$ Without
loss of generality, we assume that $h(z)=z^{t}a(z),$ where
$t\in\mathbb{N},~{}a\in\mathcal{H}(\mathbb{D}),~{}a(z)\neq 0$ and $a(0)=1.$
Further, since $h_{j}\rightarrow h$ locally uniformly in $\mathbb{D},$ we can
assume that

$h_{j}(z)=(z-z_{j,1})^{t_{1}}(z-z_{j,2})^{t_{2}}\cdots(z-z_{j,l})^{t_{l}}a_{j}(z),$

where $\sum\limits_{i=1}^{l}t_{i}=t,$ $z_{j,i}\rightarrow 0$ for $1\leq i\leq
l$ and $a_{j}(z)\rightarrow a(z)$ locally uniformly in $\mathbb{D}.$ Again, we
may assume that $z_{j,1}=0,$ since $\left\\{f_{j}(z)\right\\}$ is normal in
$\mathbb{D}$ if and only if $\left\\{f_{j}(z+z_{j,1})\right\\}$ is normal in
$\mathbb{D}$ (see [32, p. 35]). Now, we shall prove the normality of
$\left\\{f_{j}\right\\}$ at $z=0$ by applying the principle of mathematical
induction on $t.$
Note that if $t=1,$ then $l=1$ and so $h_{j}(z)=za_{j}(z).$ Thus, by
Proposition 4.7, $\left\\{f_{j}\right\\}$ is normal at $z=0.$ Also, if $l=1,$
then $h_{j}(z)=z^{t}a_{j}(z),$ and again by Proposition 4.7,
$\left\\{f_{j}\right\\}$ is normal at $z=0.$ So, let $l\geq 2$ and for
$n\in\mathbb{N}$ with $1<t<n,$ suppose that $\left\\{f_{j}\right\\}$ is normal
at $z=0.$ In accordance with the principle of mathematical induction, we only
need to show that $\left\\{f_{j}\right\\}$ is normal at $z=0$ when $n=t.$
By rearranging the zeros of $h_{j},$ if necessary, we can assume that
$|z_{j,i}|\leq|z_{j,l}|$ for $2\leq i\leq l.$ Let $z_{j,l}=w_{j}.$ Then
$w_{j}\rightarrow 0.$ Define

$g_{j}(z):=\frac{f_{j}(w_{j}z)}{w_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}\mbox{
and }v_{j}(z):=\frac{h_{j}(w_{j}z)}{w_{j}^{t}},~{}z\in
D_{r_{j}}(0),~{}r_{j}\rightarrow\infty.$

Then an easy computation shows that $Q[g_{j}](z)=w_{j}^{-t}Q[f_{j}](w_{j}z)$
and

$v_{j}(z)=z^{t_{1}}\left(z-\frac{z_{j,2}}{w_{j}}\right)^{t_{2}}\cdots\left(z-\frac{z_{j,l-1}}{w_{j}}\right)^{t_{l-1}}(z-1)^{t_{l}}a_{j}(w_{j}z)\rightarrow
v(z)$

locally uniformly in $\mathbb{C}.$ Clearly, $0$ and $1$ are two distinct zeros
of $v$ and hence all zeros of $v$ have multiplicities at most $t-1.$ Since

$Q[g_{j}](z)-v_{j}(z)=\frac{Q[f_{j}](w_{j}z)-h_{j}(w_{j}z)}{w_{j}^{t}}$ (5.1)

and $Q[f_{j}](w_{j}z)-h_{j}(w_{j}z)$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$
distinct zeros, it follows that $Q[g_{j}](z)-v_{j}(z)$ has at most
$\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros in $\mathbb{C}.$ Thus, by
induction hypothesis, we find that $\left\\{g_{j}\right\\}$ is normal in
$\mathbb{C}.$ Suppose that $g_{j}\rightarrow g$ spherically locally uniformly
in $\mathbb{C}.$ Then either $g\in\mathcal{M}(\mathbb{C})$ or $g\equiv\infty.$
Case 1: $g\in\mathcal{M}(\mathbb{C}).$
Since $g_{j}(z)\neq 0,$ it follows that either $g(z)\neq 0$ or $g\equiv 0.$
First, suppose that $g(z)\neq 0.$ Since $g_{j}\rightarrow g$ spherically
locally uniformly in $\mathbb{C},$ it follows that $Q[g_{j}]\rightarrow Q[g]$
in every compact subset of $\mathbb{C}$ disjoint from the poles of $g.$ Then
from (5.1), we find that $Q[g]-v$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$
distinct zeros in $\mathbb{C}$ and thus by Corollary 4.4 and Proposition 4.5,
$g$ has to be a constant.

Next, we claim that $\left\\{f_{j}\right\\}$ is holomorphic in
$D_{\delta/2}(0)$ for some $\delta\in(0,1).$ Suppose, on the contrary, that
$\left\\{f_{j}\right\\}$ is not holomorphic in $D_{\delta/2}(0)$ for any
$\delta\in(0,1).$ Then there exists sequence $\eta_{j}\in D_{\delta/2}(0)$
such that $\eta_{j}\rightarrow 0$ and $f_{j}(\eta_{j})=\infty.$ Assume that
$\eta_{j}$ has the smallest modulus among the poles of $f_{j}.$ It is easy to
see that $\eta_{j}/w_{j}\rightarrow\infty,$ otherwise,

$f_{j}(\eta_{j})=w_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}g_{j}(\eta_{j}/w_{j})\rightarrow
0,\mbox{ a contradiction }.$

Let

$\psi_{j}(z):=\frac{f_{j}(\eta_{j}z)}{\eta_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}\mbox{
and }u_{j}(z):=\frac{h_{j}(\eta_{j}z)}{\eta_{j}^{t}},~{}z\in
D_{r_{j}}(0),~{}r_{j}\rightarrow\infty.$

Then

$Q[\psi_{j}](z)-u_{j}(z)=\frac{Q[f_{j}](\eta_{j}z)-h_{j}(\eta_{j}z)}{\eta_{j}^{t}}$
(5.2)

and

$u_{j}(z)=z^{t_{1}}\left(z-\frac{z_{j,2}}{\eta_{j}}\right)^{t_{2}}\cdots\left(z-\frac{w_{j}}{\eta_{j}}\right)^{t_{l}}a_{j}(\eta_{j}z)\rightarrow
z^{t}$

locally uniformly in $\mathbb{C}.$ From Lemma 4.6, it follows that
$\left\\{\psi_{j}\right\\}$ is normal in
$\mathbb{C}\setminus\left\\{0\right\\}.$ Since $\psi_{j}(z)\neq 0$ and
$\psi_{j}$ is holomorphic in $\mathbb{D},$ one can easily see that
$\left\\{\psi_{j}\right\\}$ is normal in $\mathbb{D}$ and hence in
$\mathbb{C}.$ Assume that $\psi_{j}\rightarrow\psi$ spherically locally
uniformly in $\mathbb{C},$ where $\psi\in\mathcal{M}(\mathbb{C})$ or
$\psi\equiv\infty.$ Since

$\psi_{j}(0)=\frac{f_{j}(0)}{\eta_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}=\left(\frac{w_{j}}{\eta_{j}}\right)^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}g_{j}(0)\rightarrow
0,$

therefore, $\psi\not\equiv\infty.$ Also, since $\psi_{j}(z)\neq 0,$ we have
$\psi(z)\neq 0$ or $\psi\equiv 0.$ However, the latter is not possible since
$\infty=\psi_{j}(1)\rightarrow\psi(1)=\infty.$ Thus, $\psi(z)\neq 0.$ Note
that $Q[\psi_{j}](z)-u_{j}(z)\rightarrow Q[\psi](z)-z^{t}$ spherically
uniformly in every compact subset of $\mathbb{C}$ disjoint from the poles of
$\psi,$ so by (5.2), we conclude that $Q[\psi](z)-z^{t}$ has at most
$\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros in $\mathbb{C}.$ By Corollary 4.4
and Proposition 4.5, $\psi$ reduces to a constant, which contradicts the fact
that $\infty=\psi_{j}(1)\rightarrow\psi(1)=\infty.$ Hence
$\left\\{f_{j}\right\\}$ is holomorphic in $D_{\delta/2}(0).$ Since
$f_{j}(z)\neq 0,$ it follows that $\left\\{f_{j}\right\\}$ is normal at $z=0.$
Next, suppose that $g\equiv 0.$ Then by the preceding discussion, one can
easily see that $\left\\{f_{j}\right\\}$ is holomorphic in $D_{\delta/2}(0)$
and hence $\left\\{f_{j}\right\\}$ is normal at $z=0.$
Case 2: $g\equiv\infty.$
Let $\phi_{j}(z):=f_{j}(z)/z^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}.$ Then
$1/\phi_{j}(0)=0.$
Subcase 2.1: When $\left\\{1/\phi_{j}\right\\}$ is normal at $z=0.$
Then $\left\\{\phi_{j}\right\\}$ is normal at $z=0$ and so there exists $r>0$
with $D_{r}(0)\subseteq\mathbb{D}$ such that $\left\\{\phi_{j}\right\\}$ is
normal in $D_{r}(0).$ Assume that $\phi_{j}\rightarrow\phi$ spherically
locally uniformly. Since $\phi_{j}(0)=\infty,$ there exists $\rho>0$ such
that, for sufficiently large $j,$ $|\phi_{j}(z)|\geq 1$ in $D_{\rho}(0)\subset
D_{r}(0).$ Also, since $f_{j}(z)\neq 0$ in $D_{\rho}(0),$ $1/f_{j}$ is
holomorphic in $D_{\rho}(0)$ and hence

$\left|\frac{1}{f_{j}(z)}\right|=\left|\frac{1}{\phi_{j}(z)}\cdot\frac{1}{z^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}\right|\leq\left(\frac{2}{\rho}\right)^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}\mbox{
in }\partial D_{\rho/2}(0).$

Then by the maximum principle and Montel’s theorem [32, p.35], we conclude
that $\left\\{f_{j}\right\\}$ is normal at $z=0.$
Subcase 2.2: When $\left\\{1/\phi_{j}\right\\}$ is not normal at $z=0.$
By Montel’s theorem, it follows that, for every $\epsilon>0,$
$\left\\{1/\phi_{j}(z)\right\\}$ is not locally uniformly bounded in
$D_{\epsilon}(0).$ Therefore, we can find a sequence $\epsilon_{j}\rightarrow
0$ such that $1/\phi_{j}(\epsilon_{j})\rightarrow\infty.$ Since
$\left|1/\phi_{j}\right|$ is continuous, there exists $b_{j}\rightarrow$ such
that $\left|1/\phi_{j}(b_{j})\right|=1.$
Let

$K_{j}(z):=\frac{f_{j}(b_{j}z)}{b_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}},~{}z\in
D_{r_{j}}(0),~{}r_{j}\rightarrow\infty\mbox{ and
}q_{j}(z):=\frac{h_{j}(b_{j}z)}{b_{j}^{t}}.$

Then $K_{j}(z)\neq 0$ and a simple computation shows that

$Q[K_{j}](z)=\frac{Q[f_{j}](b_{j}z)}{b_{j}^{t}}\mbox{ and
}q_{j}(z)=z^{t_{1}}\left(z-\frac{z_{j,2}}{b_{j}}\right)^{t_{2}}\cdots\left(z-\frac{w_{j}}{b_{j}}\right)^{t_{l}}a_{j}(b_{j}z).$

Note that

$g_{j}\left(\frac{b_{j}}{w_{j}}\right)=\frac{f_{j}(b_{j})}{w_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}=\frac{f_{j}(b_{j})}{b_{j}^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}}\cdot\left(\frac{b_{j}}{w_{j}}\right)^{(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}}\rightarrow\infty.$

Since $|1/\phi_{j}(b_{j})|=1$ and $(t+\mu_{Q}-\lambda_{Q})/\lambda_{Q}>0,$ it
follows that $b_{j}/w_{j}\rightarrow\infty,$ and hence $w_{j}/b_{j}\rightarrow
0.$ This implies that $q_{j}(z)\rightarrow z^{t}$ locally uniformly in
$\mathbb{C}.$ Further, since
$Q[K_{j}](z)-q_{j}(z)=(Q[f_{j}](b_{j}z)-h_{j}(b_{j}z))/b_{j}^{t},$ it follows
that $Q[K_{j}]-q_{j}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros
in $\mathbb{C}$ and hence by Lemma 4.6, $\left\\{K_{j}\right\\}$ is normal in
$\mathbb{C}\setminus\left\\{0\right\\}.$ We claim that
$\left\\{K_{j}\right\\}$ is normal in $\mathbb{C}.$ Suppose otherwise. Then by
Lemma 4.2, there is a subsequence of $\left\\{K_{j}\right\\}$, which for the
sake of convenience, is again denoted by $\left\\{K_{j}\right\\}$ such that
$K_{j}(z)\rightarrow 0$ in $\mathbb{C}\setminus\left\\{0\right\\},$ which is
not possible since $|K_{j}(1)|=1.$ This establishes the claim. Now, suppose
that $K_{j}\rightarrow K$ spherically locally uniformly in $\mathbb{C}.$ It is
evident that $K(z)\neq 0$ in $\mathbb{C}$ and $K\not\equiv\infty,$ as
$K(1)=1.$ Then $Q[K_{j}]\rightarrow Q[K]$ spherically uniformly in every
compact subset of $\mathbb{C}$ disjoint from the poles of $K.$ Since
$Q[K_{j}]-q_{j}$ has at most $\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros in
$\mathbb{C},$ it follows that $Q[K]-z^{t}$ has at most
$\mu_{Q}+\lambda_{Q}(m-1)-1$ distinct zeros in $\mathbb{C}$ and so by
Corollary 4.4 and Proposition 4.5, $K$ reduces to a constant. Using the same
arguments as in Case 1, we find that $\left\\{f_{j}\right\\}$ is normal at
$z=0.$ This completes the induction process and hence the proof.

∎

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