2022

[1]Nikhil Bharti

1]Department of Mathematics, University of Jammu, Jammu, 180006, Jammu and
Kashmir, India

2]Department of Mathematics, Maulana Azad Memorial College, Jammu, 180006,
Jammu and Kashmir, India

# Normality through partial sharing of sets with differential polynomials

Kuldeep Singh Charak<EMAIL_ADDRESS><EMAIL_ADDRESS>Anil
Singh<EMAIL_ADDRESS>[  [

###### Abstract

This article aims at finding sufficient conditions for a family of meromorphic
functions to be normal by involving partial sharing of sets with differential
polynomials. Moreover, corresponding results for normal meromorphic functions
are also established which improve and generalize many known results.

###### keywords:

Normal families, normal functions, partially shared sets, meromorphic
functions, differential polynomials

###### pacs:

[

MSC Classification]Primary 30D45, 34M05; Secondary 30D30, 30D35

##  1 Introduction and Statement of Results

A family $\mathcal{F}$ of meromorphic functions in a domain
$D\subset\mathbb{C}$ is said to be normal in $D$ if every sequence of
functions in $\mathcal{F}$ contains a subsequence which converges locally
uniformly in $D$ with respect to the spherical metric to a function which is
either meromorphic in $D$ or identically $\infty.$ Normality of a family of
holomorphic functions in $D$ is defined analogously with respect to the
Euclidean metric (see schiff ; zalcman-1 ; zalcman-2 ). On the other hand, a
meromorphic function $f$ in the open unit disk $\mathbb{D}$ is said to be
normal if the family
$\mathcal{G}=\left\\{f\circ\psi:\psi\in\mathcal{A}(\mathbb{D})\right\\}$ forms
a normal family in $\mathbb{D},$ where $\mathcal{A}(\mathbb{D})$ is the group
of conformal automorphisms of $\mathbb{D}.$ The study of normal functions was
initiated implicitly by Noshiro noshiro  in $1939.$ Subsequently, Lehto and
Virtanen lehto  extended the definition of normal meromorphic functions in
$\mathbb{D}$ to arbitrary simply connected domains. Following the work of
Yoshida yoshida , Noshiro (noshiro, , Theorem 1) gave a fundamental criterion
for normal functions in terms of growth of their spherical derivatives, namely
that a meromorphic function $f$ in the open unit disk $\mathbb{D}$ is normal
if and only if $\sup\limits_{z\in\mathbb{D}}(1-\lvert
z\rvert^{2})f^{\\#}(z)<\infty.$ Owing to its immense pertinency in geometric
function theory, particularly in analyzing the boundary behaviour of a
meromorphic function, many authors have explored properties of normal
meromorphic functions from the geometric as well as the analytic point of view
(see, for example anderson ; pommerenke ; yamashita ).

For the sake of convenience, we shall denote by $\mathcal{H}(D)$ and
$\mathcal{M}(D)$ respectively the classes of all holomorphic and meromorphic
functions in a domain $D\subseteq\mathbb{C},$
$\mathbb{C}_{\infty}=\mathbb{C}\cup\left\\{\infty\right\\}$ shall denote the
extended complex plane, by $\mathbb{D}$ we shall denote the open unit disk and
finally $D(a,r)$ shall denote the open disk with center $a$ and radius $r.$

Let $f,g\in\mathcal{M}(D)$ and $S$ be a subset of $\mathbb{C}.$ Then we say
that $f$ and $g$ share the set $S$ in $D$ if $E(f,S)=E(g,S),$ where

$E(f,S):=\bigcup\limits_{s\in S}\left\\{z\in D:f(z)=s,~{}\mbox{ignoring
multiplicity}\right\\}.$

More generally, for any $S_{1},S_{2}\subseteq\mathbb{C}$ and
$f,g\in\mathcal{M}(D),$ the pair $(f,g)$ is said to share the pair
$(S_{1},S_{2})$ if $E(f,S_{1})=E(g,S_{2}).$ In this case, we write $f(z)\in
S_{1}\Leftrightarrow g(z)\in S_{2}.$ However, if either $E(f,S_{1})\subseteq
E(g,S_{2})$ or $E(f,S_{1})\supseteq E(g,S_{2}),$ then we say that $(f,g)$
share $(S_{1},S_{2})$ partially and we write $f(z)\in
S_{1}\Rightarrow(\mbox{or}\Leftarrow)~{}g(z)\in S_{2}.$ In particular, if
$f(z)=a\Leftrightarrow g(z)=a$ (respectively, $f(z)=a\Rightarrow g(z)=a$) for
some value $a\in\mathbb{C}_{\infty},$ then $a$ is said to be a shared value
(respectively, partially shared value) of $f$ and $g$ in $D.$

Our objectives here are to obtain normality criteria for a family of
meromorphic functions by involving partial sharing of sets with differential
polynomials, and to find the corresponding criteria for normal meromorphic
functions. Before we state our main results, we give some necessary
background.

Although the correspondence between normality and shared values was long
established in $1992$ by Schwick (see (schwick, , Theorem 1)), yet the first
connection between normality and shared sets was established by Liu and Pang
(liu-pang, , Theorem 1) in $2007.$ Precisely, they proved:

######  Theorem 1.

Let $\mathcal{F}\subset\mathcal{M}(D)$ and
$S=\left\\{a_{1},a_{2},a_{3}\right\\}$ be a set in $\mathbb{C}.$ If for each
$f\in\mathcal{F},$ $f(z)\in S\Leftrightarrow f^{\prime}(z)\in,$ then
$\mathcal{F}$ is normal in $D.$

Other results related to normal families and shared sets can be found in cai ;
chang-wang ; fang-zalcman ; li . It is noteworthy to mention that due to the
close relationship between normal families and normal functions, one
anticipates a criterion for normal functions corresponding to a known
criterion of normal families and vice versa. However, this correlation does
not always hold, for instance see (hayman-storvick, , p. 193).

In $2016,$ Xu and Qiu (xu-qiu, , Theorem 1.2) considered the analogous problem
of sharing of sets with derivatives for normal functions and obtained the
following:

######  Theorem 2.

Let $f\in\mathcal{M}(\mathbb{D})$ and let
$S_{1}=\left\\{a_{1},a_{2},a_{3}\right\\}$ and
$S_{2}=\left\\{b_{1},b_{2},b_{3}\right\\}$ be two set in $\mathbb{C}.$ If
$f(z)\in S_{1}\Leftrightarrow f^{\prime}(z)\in S_{2}$ in $\mathbb{D},$ then
$f$ is a normal function.

In $2019,$ Chen and Tong (chen-tong, , Theorem 1) considered sharing of sets
with higher derivatives and proved the following criterion for normal
functions:

######  Theorem 3.

Let $f\in\mathcal{M}(\mathbb{D}),$ $S_{1}=\left\\{a_{1},a_{2},a_{3}\right\\}$
and $S_{2}=\left\\{b_{1},b_{2},b_{3}\right\\}$ be two finite subsets of
$\mathbb{C}$ and $k\in\mathbb{Z}^{+}.$ If

$f(z)\in S_{1}\Leftrightarrow f^{(k)}(z)\in S_{2},$

and

$\max\limits_{0\leq i\leq k-1}\lvert f^{(i)}(z)\rvert=0\mbox{ whenever
}~{}f(z)\in S_{1}$

hold in $\mathbb{D},$ then $f$ is a normal function.

In $2020,$ Cai et al. (cai, , Theorem 2) obtained normality criterion for
families of meromorphic functions corresponding to Theorem 3 as:

######  Theorem 4.

Let $\mathcal{F}\subset\mathcal{M}(D),$
$S_{1}=\left\\{a_{1},a_{2},a_{3}\right\\}$ and
$S_{2}=\left\\{b_{1},b_{2},b_{3}\right\\}$ be two finite subsets of
$\mathbb{C}.$ Let $k$ and $m$ be two positive integers and suppose that for
each $f\in\mathcal{F}$ and $a\in S_{1},$ $f-a$ has zeros of multiplicity at
least $k.$ If

$f(z)\in S_{1}\Leftrightarrow\left(f^{(k)}\right)^{m}\in S_{2},$

then $\mathcal{F}$ is normal in $D.$

Recently Singh and Lal (vb, , Theorem 1) improved upon Theorem 3 by
considering partial sharing of sets in the following manner:

######  Theorem 5.

Let $f\in\mathcal{M}(\mathbb{D}),$ $S_{1}$ and $S_{2}$ be any two finite sets
in $\mathbb{C}$ with $\\#(S_{1})\geq 3$ and $k\in\mathbb{Z}^{+}.$ If

$f^{(k)}(z)\in S_{1}\Rightarrow f(z)\in S_{2},$

and

$\max\limits_{1\leq i\leq k}\lvert f^{(i)}(z)\rvert=0\mbox{ whenever
}~{}f(z)\in S_{1}$

in $\mathbb{D},$ then $f$ is a normal function.

Note that the condition “$\max\limits_{1\leq i\leq k}\lvert
f^{(i)}(z)\rvert=0\mbox{ whenever }~{}f(z)\in S_{1}$" in Theorem 5 is
equivalent to the condition that for each $a\in S_{1},$ $f-a$ has zeros of
multiplicity at least $k+1.$

Regarding Theorem 4 and Theorem 5, we consider the following questions:

  * Q.$1$

Does the conclusion of Theorem 4 remain valid under the weaker hypothesis of
partial sharing of sets between $f^{(k)}$ and $f$? If yes, then does the
conclusion still remain valid if $f^{(k)}$ is replaced by a differential
polynomial of $f?$

  * Q.$2$

Does the conclusion of Theorem 5 remain valid if $f^{(k)}$ is replaced by a
differential polynomial of $f?$

  * Q.$3$

Does there exist a normality criterion for a function $f\in\mathcal{M}(D)$
under the hypothesis of Theorem 4?

  * Q.$4$

Does there exist a normality criterion for a family
$\mathcal{F}\subset\mathcal{M}(D)$ under the hypothesis of Theorem 5?

To answer these questions, we need some preparation:

Let $k$ be a positive integer and let $n_{0},n_{1},\ldots,n_{k}$ be non-
negative integers (not all zeros). A mapping
$M:\mathcal{M}(D)\rightarrow\mathcal{M}(D)$ given by

$M[f]=b\cdot\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^{n_{j}}~{}\mbox{for
all}~{}f\in\mathcal{M}(D),$

where $a\in\mathcal{M}(D),~{}b\not\equiv 0,$ is called a differential monomial
of degree, $d(M):=\sum\limits_{j=0}^{k}n_{j}$ and weight,
$w(M):=\sum\limits_{j=0}^{k}(j+1)n_{j}.$

We call $b,$ the coefficient of $M$ and if $b\equiv 1,$ then $M$ is said to be
normalized differential monomial. Also, the number $k$ is known as the
differential order of $M.$

Let $f\in\mathcal{M}(D)$ and

$M_{i}[f]:=b_{i}\prod\limits_{j=0}^{k_{i}}\left(f^{(j)}\right)^{n_{j}},$

$1\leq i\leq m,$ be $m$ differential monomials. Then the sum

$P:=\sum\limits_{i=1}^{m}M_{i}$

is called a differential polynomial and the quantities
$d(P):=\max\left\\{d(M_{i}):1\leq i\leq m\right\\}$ and
$w(P):=\max\left\\{w(M_{i}):1\leq i\leq m\right\\}$ are called the degree and
weight of the differential polynomial $P$ respectively. Also, we call the
number $\kappa:=\max\left\\{k_{i}:1\leq i\leq m\right\\},$ the differential
order of $P.$ If $d(M_{1})=d(M_{2})=\cdots=d(M_{m}),$ then $P$ is said to be a
homogeneous differential polynomial.

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

$P=\sum\limits_{i=1}^{m}a_{i}~{}M_{i},$ (1.1)

where

$M_{i}[f]=\prod\limits_{j=1}^{k_{i}}\left(f^{(j)}\right)^{n_{j}},1\leq i\leq
m~{}$

are normalized differential monomials and the coefficients $a_{i}$ are
holomorphic in $D.$

Moreover, we set

$\displaystyle
1+\alpha:=\frac{w(M_{1})}{d(M_{1})}\geq\frac{w(M_{t})}{d(M_{t})},~{}\mbox{ for
}~{}2\leq t\leq m.$ (1.2)

Note that the arrangement on the right hand side of (1.2) occurs naturally.

Furthermore, we assume that the coefficients $a_{i}$ are non-vanishing for
those $M_{i}$ for which

$\frac{w(M_{i})}{d(M_{i})}=1+\alpha~{}\mbox{and}~{}\alpha\geq\kappa~{}\mbox{is
an integer}.$

Finally, we assume that the reader is familiar with the standard notations of
the Nevanlinna Theory like $m(r,f),~{}N(r,f),~{}T(r,f)$ (see hayman ).

Now we state our main results:

######  Theorem 6.

Let $\mathcal{F}\subset\mathcal{M}(D)$ and $S_{1}$ and $S_{2}$ be two finite
subsets of $\mathbb{C}$ with $\\#(S_{1})\geq 3.$ Let $P$ be a differential
polynomial defined in (1.1) and satisfying (1.2). Suppose that for each
$f\in\mathcal{F}$ and $a\in S_{1},$ $f-a$ has zeros of multiplicity at least
$\alpha+1.$ If

$P\left[f\right](z)\in S_{1}\Rightarrow f(z)\in S_{2}$

in $D,$ then $\mathcal{F}$ is normal in $D.$

######  Example 1.

Let $\mathcal{F}=\left\\{f_{n}:n\geq 2\right\\}$ be a family of meromorphic
functions in the punctured open unit disk
$\mathbb{D}^{*}:=\left\\{z\in\mathbb{C}:0<\lvert z\rvert<1\right\\}$ given by

$f_{n}(z)=\frac{nz^{2}}{2}+n.$

Then it is easy to see that $\lvert f_{n}(z)\rvert\geq 1$ in $\mathbb{D}^{*}.$
Let $S_{1}=\left\\{a_{1},a_{2},a_{3}\right\\},$ where $a_{i}$’s are distinct
complex numbers such that $\lvert a_{i}\rvert<1~{}(i=1,2,3).$ Clearly, for
each $a\in S_{1},$ $f_{n}-a$ has zeros of arbitrary multiplicity.

Let

$P[f_{n}]=f_{n}^{\prime}+\frac{1}{z^{2}}\cdot(f_{n}^{\prime})^{2}.$

Then $P[f_{n}](z)=nz+n^{2}$ and so $P[f_{n}](z)\in S_{1}\Rightarrow
f_{n}(z)\in S_{2},$ for any finite set $S_{2}$ in $\mathbb{C}.$ Obviously, the
family $\mathcal{F}$ is normal in $\mathbb{D}^{*}.$ This illustrates Theorem
6.

Theorem 6 gives affirmative answers to questions Q.$1$ and Q.$4.$

######  Remark 1.

If $\mathcal{F}$ happens to be a family of holomorphic functions in $D$ and
$0\notin S_{1},$ then the condition $\\#(S_{1})\geq 3$ in Theorem 6 can be
replaced by the condition $\\#(S_{1})\geq 2.$ This is a direct consequence of
Picard’s Theorem. In particular, if the differential polynomial $P$ in Theorem
6 is replaced by some ordinary derivative $\left(f^{(k)}\right)^{m}$ with
$\alpha$ replaced by $k,$ where $k,~{}m$ are positive integers, then the
conclusion of Theorem 6 remains valid even if $0\in S_{1}$ by (chen-fang, ,
Lemma 4). However, the cardinality of $S_{1}$ cannot be reduced to one as the
following example demonstrates:

######  Example 2.

Let $k$ and $m$ be any two positive integers and
$\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ be a family of holomorphic
functions in $\mathbb{D}$ given by

$f_{n}(z)=e^{nz}.$

Let $P[f_{n}]=\left(f_{n}^{(k)}\right)^{m}$ and $S_{1}=\left\\{0\right\\}.$
Since each $f_{n}$ omits $0,$ it follows that $f_{n}$ has zeros of arbitrary
multiplicity. Obviously, $P[f_{n}](z)\in S_{1}\Rightarrow f_{n}(z)\in S_{2},$
for any finite set $S_{2}$ in $\mathbb{C}.$ However, the family $\mathcal{F}$
is not normal in $\mathbb{D}.$

A criterion for normal meromorphic functions corresponding to Theorem 6 is

######  Theorem 7.

Let $f\in\mathcal{M}(\mathbb{D})$ and $S_{1}$ and $S_{2}$ be two finite
subsets of $\mathbb{C}$ with $\\#(S_{1})\geq 3.$ Let $P$ be a differential
polynomial defined in (1.1) and satisfying (1.2). Suppose that for each $a\in
S_{1},$ $f-a$ has zeros of multiplicity at least $\alpha+1.$ If

$P\left[f\right](z)\in S_{1}\Rightarrow f(z)\in S_{2}$

in $\mathbb{D},$ then $f$ is a normal function.

Theorem 7 gives affirmative answer to question Q.$2.$ Also, one can obtain an
affirmative answer to question Q.$3$ by a simple modification of Theorem 7
with $S_{1}\mbox{ and }S_{2}$ as three point sets and by considering the
sharing as $f(z)\in S_{1}\Leftrightarrow\left(f^{(k)}\right)^{m}\in S_{2}$
under the assumption that for each $a\in S_{1},$ $f-a$ has zeros of
multiplicity at least $k.$

######  Remark 2.

It is important to note that if the cardinality of the set $S_{1}$ in Theorem
6 and Theorem 7 is greater than $4$, then the condition “for each $a\in
S_{1},$ $f-a$ has zeros of multiplicity at least $\alpha+1$" is enough to
ensure normality. This follows immediately by a simple application of Lemma 13
and Lemma 12, respectively, together with the fact that a non-constant
meromorphic function cannot have more than four totally ramified values.
Furthermore, the set $S_{1}$ in Theorem 6 and Theorem 7 can be taken to be a
two point set consisting of distinct non-zero complex values of which at least
one is not a Picard exceptional value. Also, it easily follows from (grahl, ,
Theorem 5) that the two point set $S_{1}$ may contain zero if the differential
polynomial $P$ is of the form $P=\sum\limits_{i=1}^{m}a_{i}M_{i}$ with non-
zero constant coefficients $a_{i}$ and normalized differential monomials
$M_{i}$ in $f^{\prime}$ satisfying at least one of the following conditions:

  * $(i)$

$d(M_{i})\geq 2$ for all $i=1,2,\ldots,m\mathchar 24635\relax\;$

  * $(ii)$

$w(M_{1})\geq w(M_{j})+2$ for all $j=2,\ldots,m.$

The following example shows that the condition “for each $f\in\mathcal{F}$ and
$a\in S_{1},$ $f-a$ has zeros of multiplicity at least $\alpha+1$" in Theorem
6 cannot be dropped.

######  Example 3.

Let $\mathcal{F}=\left\\{f_{n}:f_{n}(z)=nz,~{}n\in\mathbb{N}\right\\}$ be a
family of meromorphic functions in $\mathbb{D}$ and let $S_{1}$ be any three
point set in $\mathbb{C}\setminus\mathbb{N}.$ Then for each $a\in S_{1},$
$f_{n}-a$ has only simple zeros. Consider $P[f_{n}]=f_{n}^{\prime}.$ Then
$P[f_{n}](z)=n.$ Clearly,

$P[f_{n}](z)\in S_{1}\Rightarrow f_{n}(z)\in S_{2}$

for any finite set $S_{2}$ in $\mathbb{C}.$ However, the family $\mathcal{F}$
is not normal in $\mathbb{D}.$

The next example demonstrates that the condition “$P\left[f\right](z)\in
S_{1}\Rightarrow f(z)\in S_{2}$" in Theorem 6 is essential.

######  Example 4.

Let $\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ be a family of
holomorphic functions in $\mathbb{D}$ given by

$f_{n}(z)=\cos\left(e^{z+n}\right).$

Let $S_{1}=S_{2}=\left\\{-1,1\right\\}$ and $P[f_{n}]=f_{n}^{\prime}.$ Then
for each $a\in S_{1},$ $f_{n}-a$ has zeros of multiplicity $2$ and
$P[f_{n}](z)\in S_{1}\not\Rightarrow f_{n}(z)\in S_{2}.$ However, the family
$\mathcal{F}$ is not normal in $\mathbb{D}$ since along real axis, the family
$\mathcal{F}$ is uniformly bounded by $1$ but along imaginary axis, the limit
function is $\infty.$

The condition “for each $a\in S_{1},$ $f-a$ has zeros of multiplicity at least
$\alpha+1$" in Theorem 7 is not redundant as demonstrated by the following
example (one may also see (hayman-storvick, , p.193)):

######  Example 5.

Consider

$f(z)=2(1-z)~{}\mbox{exp}\left\\{\frac{2+z}{1-z}\right\\}$

in the open unit disk $\mathbb{D}.$ Then

$f^{\prime}(z)=\frac{4+2z}{1-z}~{}\mbox{exp}\left\\{\frac{2+z}{1-z}\right\\}$

and $f(z)\neq 0$ in $\mathbb{D}.$ Also, one can easily see that $\lvert
f^{\prime}(z)\rvert>\sqrt{e}$ in $\mathbb{D}.$ Let
$S_{1}=\left\\{0,a_{1},a_{2}\right\\},$ where $a_{i}$’s $(i=1,2)$ are non-zero
distinct complex numbers with $\lvert a_{i}\rvert<\sqrt{e}$ and $S_{2}$ be any
finite set in $\mathbb{C}.$ Note that $f-a_{i}$ does not have zeros of
multiplicity at least $2.$ Let $P[f]=f^{\prime}$ and $a_{3}$ be any non-zero
complex number distinct from $a_{1}$ and $a_{2}$ such that $\lvert
a_{3}\rvert<\sqrt{e}.$ Clearly, $\left\lvert
f^{\prime\prime}(z)\right\rvert\leq M$ whenever $f^{\prime}(z)=a_{3}$ for any
$M>0$ and $P\left[f\right](z)\in S_{1}\Rightarrow f(z)\in S_{2}.$ However, $f$
is not a normal function since if

$z=\frac{1}{2}(1+e^{i\theta}),~{}0<\theta<2\pi,$ $\mbox{then}~{}~{}~{}\lvert
f(z)\rvert=e^{2}\lvert\sin\left(\theta/2\right)\rvert\longrightarrow
0,~{}\mbox{as}~{}\theta\longrightarrow 0,$

whereas

$f(z)\longrightarrow\infty,~{}\mbox{as}~{}z\longrightarrow
1^{-}~{}\mbox{through real values}.$

The following two results establish that the cardinality of the set $S_{1}$ in
Theorem 6 and Theorem 7 can be reduced by one under suitable conditions.

######  Theorem 8.

Let $\mathcal{F}\subset\mathcal{M}(D),$ $S_{1}$ and $S_{2}$ be two finite
subsets of $\mathbb{C}$ with $S_{1}=\left\\{a_{1},a_{2}\right\\},$ where
$a_{1},~{}a_{2}$ are non-zero distinct complex numbers. Let $P$ be a
differential polynomial defined in (1.1) and satisfying (1.2). Suppose that
for each $f\in\mathcal{F},$ $f-a_{i}~{}(i=1,2)$ has zeros of multiplicity at
least $\alpha+1.$ If

$P\left[f\right](z)\in S_{1}\Rightarrow f(z)\in S_{2}$

in $D$ and if there exist some $M>0$ and a point $a_{3}\in\mathbb{C}\setminus
S_{1}$ such that $\lvert f^{\prime}(z)\rvert\leq M$ whenever $f(z)=a_{3},$
then $\mathcal{F}$ is normal in $D.$

By Remark $1,$ it follows that $S_{1}$ may contain zero if the differential
polynomial $P$ in Theorem 8 is replaced by $\left(f^{(k)}\right)^{m}$ and
$\alpha$ replaced by $k,$ where $k,~{}m$ are positive integers. However, we
cannot take $a_{1}=a_{2}$ as the following example demonstrates:

######  Example 6.

Let $a$ be any non-zero complex number and let
$\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ be a family of meromorphic
functions in $\mathbb{D}$ given by

$f_{n}(z)=\frac{a}{a+e^{nz}}$

and let $S_{1}=\left\\{0\right\\}.$ Then $P[f_{n}](z)=f_{n}^{\prime}(z)\in
S_{1}\Rightarrow f_{n}(z)\in S_{2}$ for any finite set
$S_{2}\subset\mathbb{C}$ and $f_{n}$ has zeros of arbitrary multiplicity.
Also, we have $\lvert f_{n}^{\prime}(z)\rvert\leq M$ whenever $f_{n}(z)=1$ for
any positive constant $M.$ However, the family $\mathcal{F}$ is not normal in
$\mathbb{D}.$

######  Theorem 9.

Let $f\in\mathcal{M}(\mathbb{D}),$ $S_{1}$ and $S_{2}$ be two finite subsets
of $\mathbb{C}$ with $S_{1}=\left\\{a_{1},a_{2}\right\\},$ where
$a_{1},~{}a_{2}$ are non-zero distinct complex numbers. Let $P$ be a
differential polynomial defined in (1.1) and satisfying (1.2). Suppose that
$f-a_{i}~{}(i=1,2)$ has zeros of multiplicity at least $\alpha+1.$ If

$P\left[f\right](z)\in S_{1}\Rightarrow f(z)\in S_{2}$

in $\mathbb{D}$ and if there exist some $M>0$ and a point
$a_{3}\in\mathbb{C}\setminus S_{1}$ such that $\lvert f^{\prime}(z)\rvert\leq
M$ whenever $f(z)=a_{3},$ then $f$ is a normal function.

Finally, we have another criterion for normal meromorphic functions and is of
independent interest.

######  Theorem 10.

Let $f\in\mathcal{M}(\mathbb{D}),$ $S_{1}=\left\\{a_{1},a_{2},a_{3}\right\\}$
and $S_{2}=\left\\{b_{1},b_{2},b_{3}\right\\}$ be two finite subsets of
$\mathbb{C}.$ Let $l,m\in\mathbb{N}$ and $a$ be any fixed complex number. Then
$f$ is a normal function if any one of the following holds:

  * $(i)$

$\left(f^{(l)}\right)^{m}(z)\in S_{1}\Leftrightarrow f(z)\in S_{2},$ and $f-a$
has zeros and poles of multiplicity at least $l+1,$ if $a\in S_{1}.$

  * $(ii)$

$f(z)\in S_{1}\Rightarrow\left(f^{(l)}\right)^{m}(z)\in S_{2},$ and $f-a$ has
zeros and poles of multiplicity at least $l+2,$ if $a\notin S_{1}.$

##  2 Auxiliary Results

In this section, we describe some preliminary results that are crucial to
prove main results of this paper. First recall that if
$f\in\mathcal{M}(\mathbb{C})$ and $a\in\mathbb{C}_{\infty},$ then $a$ is said
to be a totally ramified value of $f$ if $f-a$ has no simple zeros. Nevanlinna
(see (bergweiler, , p. 84)) proved the following widely known result
concerning multiplicities of $a$-points of a meromorphic function. This result
plays a major role in the proofs of Theorem 8 and Theorem 9:

######  Lemma 11 (Nevanlinna’s Theorem).

Let $f$ be a non constant meromorphic function,
$a_{1},a_{2},\ldots,a_{q}\in\mathbb{C}_{\infty}$ and
$m_{1},m_{2},\ldots,m_{q}\in\mathbb{N}.$ Suppose that all $a_{j}$-points of
$f$ have multiplicity at least $m_{j},$ for $j=1,2,\ldots,q.$ Then

$\sum\limits_{j=1}^{q}\left(1-\frac{1}{m_{j}}\right)\leq 2.$

If $f$ omits the value $a_{j}$, then $m_{j}=\infty.$

We need the following rescaling lemma due to Lohwater and Pommerenke
(lohwater, , Theorem 1):

######  Lemma 12.

Let $f\in\mathcal{M}(\mathbb{D}).$ Suppose that $f$ is not a normal function.
Then there exist points $z_{n}\in\mathbb{D}$ and positive numbers
$\rho_{n}\longrightarrow 0$ such that

$g_{n}(\zeta)=f(z_{n}+\rho_{n}\zeta)\longrightarrow g(\zeta)$

locally uniformly in $\mathbb{C}$ with respect to the spherical metric, where
$g$ is a non-constant meromorphic function on $\mathbb{C}.$

The proofs of main results in this paper rely essentially on the following
extension of the famous Zalcman-Pang Lemma due to Chen and Gu chen-gu  (see
also (zalcman-2, , p. 216), cf. (pang-zalcman, , Lemma 2)).

######  Lemma 13 (Zalcman-Pang Lemma).

Let $\mathcal{F}\subset\mathcal{M}(D)$ be a family of meromorphic functions
all of whose zeros have multiplicities at least $m$ and whose poles have
multiplicities at least $p.$ Let $-p<\alpha<m.$ If $\mathcal{F}$ is not normal
at $z_{0}\in D,$ then there exist sequences
$\left\\{f_{n}\right\\}\subset\mathcal{F},$ $\left\\{z_{n}\right\\}\subset D$
satisfying $z_{n}\longrightarrow z_{0}$ and positive numbers $\rho_{n}$ with
$\rho_{n}\longrightarrow 0$ such that the sequence $\left\\{g_{n}\right\\}$
defined by

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

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
$\zeta\in\mathbb{C},$ $g^{\\#}(\zeta)\leq g^{\\#}(0)=1.$

The following lemma due to Doeringer (doeringer, , Lemma 1 (i)) is an
extension of the Nevanlinna’s well known Lemma on the Logarithmic Derivative
(hayman, , Lemma 2.3).

######  Lemma 14.

Let $f\in\mathcal{M}(\mathbb{C})$ and $P[f]$ be a differential polynomial of
$f$ with meromorphic coefficients $a_{j},~{}j=1,\ldots,k.$ Then

$m\left(r,P[f]\right)\leq d(Q)\cdot
m(r,f)+\sum\limits_{j=1}^{k}m(r,a_{j})+S(r,f).$

We also need the following lemma in the proofs of Theorem 6 and Theorem 7.

######  Lemma 15.

Let $f\in\mathcal{M}(\mathbb{C})$ be such that $f$ has finitely many zeros and
let $P[f]$ be a differential polynomial of $f$ with coefficients
$a_{j},~{}j=1,\ldots,k$ such that $T(r,a_{j})=S(r,f).$ If $P[f]$ is constant,
then either $f$ is a polynomial or $P[f]\equiv 0.$

Proof: Since $f$ has finitely many zeros, write $f=g/h,$ for some polynomial
$g$ and some entire function $h.$ Then we have

$f^{(k)}=\frac{Q_{k}[h]}{h^{k+1}},$

where $Q_{k}[h]$ is a homogeneous differential polynomial of degree $k$ with
polynomial coefficients and

$M_{j}[f]=\frac{H_{j}[h]}{h^{w(M_{j})}}$

for $j=1,2,\ldots,k,$ where $H_{j}[h]$ are homogeneous differential
polynomials such that $d(H_{j})=w(M_{j})-d(M_{j}).$ This gives

$P[f]=\frac{1}{h^{w(P)}}\cdot\left(\sum\limits_{j=1}^{k}a_{j}h^{w(P)-w(M_{j})}H_{j}[h]\right)=\frac{U[h]}{h^{w(P)}},$

where $U[h]$ is a differential polynomial of $h$ whose coefficients are the
product of polynomials with small functions of $f.$ Moreover

$d(U)\leq\max\limits_{1\leq j\leq k}\left\\{w(P)-d(M_{j})\right\\}\leq
w(P)-1.$

Now suppose that $P[f]\equiv c$ for some non-zero constant $c.$ Then we can
write

$h^{w(P)}=\frac{1}{c}\cdot U[h].$ (2.1)

From Lemma 14, we deduce that

$w(P)\cdot m(r,h)=m(r,U[h])+\mathcal{O}(1)\leq(w(P)-1)\cdot
m(r,h)+\mathcal{O}(\log r)+S(r,f),$

showing that $T(r,h)=m(r,h)=S(r,f)=S(r,h).$ Therefore, $h$ must be a
polynomial and so is $h^{w(P)}.$ But degree of $h^{w(P)}$ is $w(P)\cdot d(h)$
and $U[h]$ is a polynomial of degree at most $\left(w(P)-1\right)\cdot d(h).$
This together with (2.1) yield $d(h)\leq 0.$ Thus $h$ is constant and so $f$
is a polynomial.

The proofs of Theorem 6 and Theorem 7 also require the following result due to
Grahl (grahl, , Lemma 13).

######  Lemma 16.

Let $f\in\mathcal{H}(\mathbb{C})$ be a non-constant function and let $P$ be a
differential polynomial such that $P[f]\equiv 0.$ Assume that $T(r,a)=S(r,f)$
for each coefficient $a$ of $P[f].$ If

$P=Q_{0}+\ldots+Q_{d}$

with homogeneous differential polynomials $Q_{j}$ of degree $j$ or
($Q_{j}\equiv 0$) and $Q_{l}[f]\not\equiv 0$ for some $l$ in the set
$\left\\{0,\ldots,d-1\right\\},$ then

$m\left(r,\frac{1}{f}\right)\leq l\cdot N\left(r,\frac{1}{f}\right)+S(r,f).$

The following value distribution result plays an important role in the proof
of Theorem 10:

######  Lemma 17.

Let $f\in\mathcal{M}(\mathbb{C}),$ $f\not\equiv$ constant, $l,q\in\mathbb{N}$
and let $S=\left\\{a_{1},a_{2},\ldots,a_{q}\right\\}$ be any finite set in
$\mathbb{C}$ such that

$f(z)\in S\Rightarrow f^{(l)}(z)=0.$

Then

$q~{}T(r,f)<(l+1)\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{f}\right)+N\left(r,\frac{1}{f}\right)+S(r,f).$

We shall prove Lemma 17 by using the following value distribution result due
to Yi (yi, , Lemma 3):

######  Lemma 18.

Let $f$ be a non-constant meromorphic function in $\mathbb{C}$ and let $l$ be
a non-negative integer. Then

$N\left(r,\frac{1}{f^{(l)}}\right)<N\left(r,\frac{1}{f}\right)+l\bar{N}\left(r,f\right)+S\left(r,f\right).$

Proof: [Proof of Lemma 17] By the Second Fundamental Theorem of Nevanlinna, we
have

$q~{}T(r,f)\leq\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{f}\right)+\sum\limits_{i=1}^{q}\overline{N}\left(r,\frac{1}{f-a_{i}}\right)+S(r,f).$

Since

$f(z)\in S\Rightarrow f^{(l)}(z)=0,$

we find that

$\sum\limits_{i=1}^{q}\overline{N}\left(r,\frac{1}{f-a_{i}}\right)\leq\overline{N}\left(r,\frac{1}{f^{(l)}}\right)$

and hence we obtain

$\displaystyle q~{}T(r,f)$ $\displaystyle\leq$
$\displaystyle\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{f}\right)+\sum\limits_{i=1}^{q}\overline{N}\left(r,\frac{1}{f-a_{i}}\right)+S(r,f)$
$\displaystyle\leq$
$\displaystyle\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{f^{(l)}}\right)+S(r,f)$
$\displaystyle\leq$
$\displaystyle\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{f}\right)+N\left(r,\frac{1}{f^{(l)}}\right)+S(r,f)$

By Lemmaa 18, we have

$N\left(r,\frac{1}{f^{(l)}}\right)<N\left(r,\frac{1}{f}\right)+l\bar{N}\left(r,f\right)+S\left(r,f\right)$

and therefore

$\displaystyle q~{}T(r,f)$ $\displaystyle<$
$\displaystyle\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{f}\right)+N\left(r,\frac{1}{f}\right)+l\bar{N}\left(r,f\right)+S(r,f)$
$\displaystyle\leq$
$\displaystyle(l+1)\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{f}\right)+N\left(r,\frac{1}{f}\right)+S(r,f).$

##  3 Proofs of Main Results

Proof: [Proof of Theorem 6] Suppose that $\mathcal{F}$ is not normal at
$z_{0}\in D.$ Then by Lemma 13, there exist sequences
$\left\\{f_{n}\right\\}\subset\mathcal{F},$ $\left\\{z_{n}\right\\}\subset D$
with $z_{n}\longrightarrow z_{0}$ and positive numbers $\rho_{n}$ satisfying
$\rho_{n}\longrightarrow 0$ such that

$g_{n}(\zeta):=f_{n}(z_{n}+\rho_{n}\zeta)\longrightarrow g(\zeta)$

locally uniformly in $\mathbb{C}$ with respect to the spherical metric, where
$g\in\mathcal{M}(\mathbb{C})$ is non-constant. By Picard’s Theorem, it follows
that $g$ assumes at least one of the values of $S_{1}.$ We claim that for any
$a\in S_{1},$ all zeros of $g-a$ have multiplicity at least $\alpha+1.$
Indeed, let $g(\zeta_{0})=a.$ Since $g\not\equiv a,$ by Hurwitz’s Theorem,
there exists $\zeta_{n}\longrightarrow\zeta_{0}$ such that for sufficiently
large $n,$

$g_{n}(\zeta_{n})=f_{n}(z_{n}+\rho_{n}\zeta_{n})=a\in S_{1}.$

By hypothesis, we have $f_{n}^{(i)}(z_{n}+\rho_{n}\zeta_{n})=0$ for $1\leq
i\leq\alpha.$

Thus

$g^{(i)}(\zeta_{0})=\lim\limits_{n\to\infty}g_{n}^{(i)}(\zeta_{n})=\lim\limits_{n\to\infty}\rho_{n}^{i}f_{n}^{(i)}(z_{n}+\rho_{n}\zeta_{n})=0$

for $1\leq i\leq\alpha,$ and this establishes the claim.

Now suppose that $\zeta_{0}$ is a zero of $g-a$ with multiplicity $l.$ Then
there exist some $\delta>0$ such that for sufficiently large $n,$ $g_{n}$ is
holomorphic in the disk $D(\zeta_{0},\delta).$ Let

$h_{n}(\zeta):=\frac{g_{n}(\zeta)-a}{\rho_{n}^{\alpha}}.$

Then $h_{n}$ is holomorphic in $D(\zeta_{0},\delta)$ and
$h_{n}(\zeta^{\prime})=0$ if and only if $g_{n}(\zeta^{\prime})=a$ and so
$h_{n}$ has zeros of multiplicity at least $\alpha+1.$

Next, we claim that $\left\\{h_{n}\right\\}$ is not normal at $\zeta_{0}.$
Suppose otherwise. Then there exist a $\delta_{1}$ such that
$0<\delta_{1}<\delta$ and a subsequence of $\left\\{h_{n}\right\\}$ (again
denoted by $\left\\{h_{n}\right\\}$) such that $h_{n}\longrightarrow h$
locally uniformly in $D(\zeta_{0},\delta_{1}),$ where $h$ is either
holomorphic or identically $\infty$ in $D(\zeta_{0},\delta_{1}).$ Since
$g(\zeta_{0})=a$ and $g\not\equiv a,$ by Hurwitz’s Theorem, we find that
$h(\zeta_{0})=0.$ Also, since zeros of $g-a$ in $D(\zeta_{0},\delta_{1})$ are
isolated, there is some $\zeta_{1}\neq\zeta_{0}$ in $D(\zeta_{0},\delta_{1})$
such that $g(\zeta_{1})\neq a.$ Thus for sufficiently large $n,$ $\lvert
g_{n}(\zeta_{1})-a\rvert>0$ and hence

$\lvert h_{n}(\zeta_{1})\rvert=\frac{\lvert
g_{n}(\zeta_{1})-a\rvert}{\rho_{n}^{\alpha}}\longrightarrow\infty.$

This implies that $\left\\{h_{n}\right\\}$ converges uniformly to $\infty$ in
$D(\zeta_{0},\delta_{1})$ which is not the case.

Again, by Lemma 13, there exists a subsequence of $\left\\{h_{n}\right\\}$
(again denoted by $\left\\{h_{n}\right\\}$), a sequence of points
$\hat{\zeta_{n}}\longrightarrow\zeta_{0}$ and positive numbers
$r_{n}\longrightarrow 0$ such that

$\phi_{n}(\xi)=\frac{h_{n}(\hat{\zeta_{n}}+r_{n}\xi)}{r_{n}^{\alpha}}=\frac{g_{n}(\hat{\zeta_{n}}+r_{n}\xi)-a}{(\rho_{n}r_{n})^{\alpha}}=\frac{f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)-a}{(\rho_{n}r_{n})^{\alpha}}$

converges locally uniformly to a non-constant entire function $\phi(\xi).$
Since $g-a$ has zeros of multiplicity at least $\alpha+1,$ it easily follows
that zeros of $\phi$ have multiplicity at least $\alpha+1.$

Claim 1: $\phi$ has finitely many zeros.
It is sufficient to show that $\phi$ has at most $l$ distinct zeros. Suppose
on the contrary that $\phi$ has $l+1$ distinct zeros, say
$\xi_{1},\xi_{2},\ldots,\xi_{l+1}.$ Then by Hurwitz’s Theorem, there exist
$l+1$ distinct sequences $\left\\{\xi_{n_{j}}\right\\}$ such that
$\xi_{n_{j}}\longrightarrow\xi_{j}$ and $\phi_{n}\left(\xi_{n_{j}}\right)=0$
for $j=1,2,\ldots,l+1.$ This implies that

$g_{n}(\hat{\zeta_{n}}+r_{n}\xi_{n_{j}})=a,~{}j=1,2,\ldots,l+1.$

Since $\hat{\zeta_{n}}+r_{n}\xi_{n_{j}}\longrightarrow\zeta_{0}$ and
$\hat{\zeta_{n}}+r_{n}\xi_{n_{i}}\neq\hat{\zeta_{n}}+r_{n}\xi_{n_{j}}$ for
$i\neq j,$ it follows that $\zeta_{0}$ is a zero of $g-a$ with multiplicity at
least $l+1$ which contradicts the fact that $\zeta_{0}$ is a zero of $g-a$
with multiplicity $l.$ This proves the claim.

Claim 2: If $P[\phi]$ assumes a non-zero finite value, say $\beta,$ then
$S_{\beta}=\left\\{f_{n}(z):P[f_{n}](z)=\beta,z\in D\right\\}$ is an infinite
set.
First note that

$\displaystyle\tilde{P}[\phi_{n}](\xi)$ $\displaystyle:=$ $\displaystyle
P[f_{n}](z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}a_{i}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)~{}\rho_{n}^{\left[(1+\alpha)d(M_{i})-w(M_{i})\right]}~{}M_{i}[\phi_{n}](\xi).$

Since

$1+\alpha=\frac{w(M_{1})}{d(M_{1})}~{}\mbox{ and
}~{}\frac{w(M_{1})}{d(M_{1})}\geq\frac{w(M_{t})}{d(M_{t})}~{}\mbox{ for
}~{}2\leq t\leq m,$

we assume, without loss of generality, that

$1+\alpha=\frac{w(M_{1})}{d(M_{1})}=\frac{w(M_{2})}{d(M_{2})}=\cdots=\frac{w(M_{s})}{d(M_{s})}$

and

$\frac{w(M_{1})}{d(M_{1})}>\frac{w(M_{t})}{d(M_{t})}~{}\mbox{for}~{}s+1\leq
t\leq m.$

Therefore, we obtain

$\displaystyle\tilde{P}[\phi_{n}](\xi)$ $\displaystyle=$ $\displaystyle
P[f_{n}](z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{s}a_{i}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)M_{i}[\phi_{n}](\xi)$
$\displaystyle+\sum\limits_{i=s+1}^{m}a_{i}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)~{}\rho_{n}^{\left[(1+\alpha)d(M_{i})-w(M_{i})\right]}~{}M_{i}[\phi_{n}](\xi)$
$\displaystyle=$
$\displaystyle\tilde{Q}[\phi_{n}](\xi)+\sum\limits_{i=s+1}^{m}a_{i}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)~{}\rho_{n}^{\left[(1+\alpha)d(M_{i})-w(M_{i})\right]}M_{i}[\phi_{n}](\xi),$

where

$\tilde{Q}[\phi_{n}](\xi):=\sum\limits_{i=1}^{s}a_{i}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)M_{i}[\phi_{n}](\xi).$

Again, since all $a_{i}~{}(1\leq i\leq m)$ are holomorphic functions in $D$
and $a_{i}(z)\neq 0$ for $1\leq i\leq s,$ it follows that

$\sum\limits_{i=s+1}^{m}a_{i}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)~{}\rho_{n}^{\left[(1+\alpha)d(M_{i})-w(M_{i})\right]}~{}M_{i}[\phi_{n}](\xi)$

converges uniformly to $0$ on compact subsets of $\mathbb{C}$ and hence

$\tilde{P}[\phi_{n}](\xi)=P[f_{n}](z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)\longrightarrow
Q[\phi](\xi)$

uniformly on compact subsets of $\mathbb{C},$ where

$Q[\phi](\xi)=\sum\limits_{i=1}^{s}a_{i}(z_{0})M_{i}[\phi](\xi).$

We claim that $Q[\phi]\not\equiv$ constant. If $Q[\phi]\equiv$ constant, then
by Lemma 15, we find that either $\phi$ is polynomial or $Q[\phi]\equiv 0.$ If
$\phi$ is a polynomial, then by Weierstrass’s Theorem, we have

$\phi_{n}^{(\alpha)}(\xi)=f_{n}^{(\alpha)}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)\longrightarrow\phi^{(\alpha)}(\xi)$

uniformly on compact subsets of $\mathbb{C}.$ Since $\phi$ is a non-constant
polynomial having zeros of multiplicity at least $\alpha+1,$ it follows that
$\phi^{(\alpha)}$ assumes every value in $\mathbb{C}.$ In particular,
$\phi^{(\alpha)}$ assumes a non-zero value $b\in S_{1}.$ Let
$\phi^{(\alpha)}(\xi^{*})=b.$ Since $\phi^{(\alpha)}(\xi)\not\equiv b,$ by
Hurwitz’s Theorem, there exists $\xi_{n}^{*}\longrightarrow\xi^{*}$ such that
for sufficiently large $n,$

$\phi_{n}^{(\alpha)}(\xi_{n}^{*})=f_{n}^{(\alpha)}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n}^{*})=b~{}(\neq
0).$

This implies that
$f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n}^{*})\neq a,$ since
if $f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n}^{*})=a,$ then by
hypothesis, we have
$f_{n}^{(\alpha)}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n}^{*})=0,$
a contradiction to the fact that
$f_{n}^{(\alpha)}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n}^{*})=b\neq
0.$

Then

$\phi(\xi^{*})=\lim\limits_{n\to\infty}\phi_{n}(\xi_{n}^{*})=\lim\limits_{n\to\infty}\frac{f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n}^{*})-a}{(\rho_{n}r_{n})^{\alpha}}=\infty,$

showing that $\xi^{*}$ is a pole of $\phi$ which is not possible. Thus $\phi$
cannot be a polynomial.

Next, if $Q[\phi]\equiv 0,$ then by Lemma 16, we deduce that

$m(r,\frac{1}{\phi})=S(r,\phi)$

and hence

$m(r,\phi)=S(r,\phi)$

showing that $\phi$ is a polynomial, which is not the case. Thus
$Q[\phi]\not\equiv 0$ and this proves our claim. Then by Picard’s Theorem,
$Q[\phi](\xi)=\beta$ has at least one solution for any $\beta\in\mathbb{C}$
with at most one exception.

Now, let $\xi_{0}\in\mathbb{C}$ be such that
$Q[\phi](\xi_{0})=\beta_{1}~{}(\neq 0)$ and $\phi(\xi_{0})=\beta_{2},$ where
$\beta_{1},\beta_{2}\in\mathbb{C}.$ It is clear that $\beta_{2}\neq 0,$
otherwise $Q[\phi](\xi_{0})=0$ since zeros of $\phi$ have multiplicity at
least $\alpha+1.$

Since $Q[\phi]\not\equiv\beta_{1},$ by Hurwitz’s Theorem, there exists
$\xi_{n}\longrightarrow\xi_{0}$ such that for sufficiently large $n,$

$P[f_{n}](z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=\beta_{1}~{}(\neq
0).$

Also, $\phi_{n}(\xi_{n})\longrightarrow\phi(\xi_{0})=\beta_{2}~{}(\neq 0)$
implies that
$f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})\longrightarrow a.$
Further, note that if
$f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=a,$ then by
hypothesis,

$f_{n}^{(i)}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=0~{}\mbox{for}~{}1\leq
i\leq\alpha$

so that

$P[f_{n}](z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=0,$

a contradiction to the fact that
$P[f_{n}](z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=\beta_{1}\neq
0.$ Therefore, $f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})\neq
a$ and hence

$S_{\beta_{1}}=\left\\{f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n}):P[f_{n}](z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=\beta_{1}\right\\}$

is an infinite set. This proves Claim $2.$

Now, since $\\#(S_{1})\geq 3,$ $S_{1}$ contains a non-zero finite value, say
$\nu.$ Then by Claim $2,$ the set $S_{\nu}$ is an infinite set and cannot be
contained in the finite set $S_{2},$ a contradiction to our hypothesis. Hence
$\mathcal{F}$ is normal in $D.$

Proof: [Proof of Theorem 7] Suppose that $f$ is not a normal function. Then by
Lemma 12, there exist points $z_{n}\in\mathbb{D}$ and positive numbers
$\rho_{n}\longrightarrow 0$ such that

$g_{n}(\zeta):=f(z_{n}+\rho_{n}\zeta)$

converges spherically locally uniformly in $\mathbb{C}$ to a non-constant
meromorphic function $g(\zeta).$ Rest of the proof goes on the lines of the
proof of Theorem 6 with simple modifications, hence omitted.

Proof: [Proof of Theorem 8] Suppose that $\mathcal{F}$ is not normal at
$z_{0}\in D.$ Then by Lemma 13, there exist sequences
$\left\\{f_{n}\right\\}\subset\mathcal{F},$ $\left\\{z_{n}\right\\}\subset D$
with $z_{n}\longrightarrow z_{0}$ and positive numbers $\rho_{n}$ satisfying
$\rho_{n}\longrightarrow 0$ such that

$g_{n}(\zeta):=f_{n}(z_{n}+\rho_{n}\zeta)\longrightarrow g(\zeta)$

locally uniformly in $\mathbb{C}$ with respect to the spherical metric, where
$g\in\mathcal{M}(\mathbb{C})$ is non-constant.

We claim that $g$ assumes at least one value from $S_{1}.$ Suppose on the
contrary that $g$ omits both $a_{1}$ and $a_{2}.$ Then by Picard’s Theorem,
$g$ assumes every value in $\mathbb{C}\setminus S_{1}.$ In particular, $g$
must assume the value $a_{3}.$ Suppose there exists $\zeta_{0}\in\mathbb{C}$
such that $g(\zeta_{0})=a_{3}.$ Since $g\not\equiv a_{3},$ by Hurwitz’s
Theorem, there exists a sequence $\zeta_{n}\longrightarrow\zeta_{0}$ such that
for sufficiently large $n,$

$g_{n}(\zeta_{n})=f_{n}(z_{n}+\rho_{n}\zeta_{n})=a_{3}.$

By hypothesis, we have $\lvert
f_{n}^{\prime}(z_{n}+\rho_{n}\zeta_{n})\rvert\leq M.$

Then

$\displaystyle\left\lvert g^{\prime}(\zeta_{0})\right\rvert$ $\displaystyle=$
$\displaystyle\left\lvert\lim\limits_{n\to\infty}g_{n}^{\prime}(\zeta_{n})\right\rvert$
$\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\left\lvert\rho_{n}f_{n}^{\prime}(z_{n}+\rho_{n}\zeta_{n})\right\rvert$
$\displaystyle\leq$ $\displaystyle\lim\limits_{n\to\infty}\rho_{n}~{}M$
$\displaystyle=$ $\displaystyle 0.$

This shows that $\zeta_{0}$ is a zero of $g-a_{3}$ of multiplicity at least
$2.$ Let $m_{1},~{}m_{2}$ and $m_{3}$ denote the multiplicity of zeros of
$g-a_{1},~{}g-a_{2}$ and $g-a_{3}$ respectively. Then by simple calculation,
we find that

$\sum\limits_{j=1}^{3}\left(1-\frac{1}{m_{j}}\right)>2,$

which is not true by Lemma 11. Now the proof is completed by following the
proof of Theorem 6.

The proof of Theorem 9 follows from the proof of Theorem 8 with minor
modifications and hence omitted.

Proof: [Proof of Theorem 10] $(i)$ Suppose that $f$ is not a normal function.
Then the function $g=f-a$ is not normal. So, by Lemma 12, there exist points
$z_{n}\in\mathbb{D}$ and numbers $\rho_{n}\longrightarrow 0$ such that

$h_{n}(\zeta):=g(z_{n}+\rho_{n}\zeta)\longrightarrow h(\zeta)$

spherically locally uniformly in $\mathbb{C},$ where
$h\in\mathcal{M}(\mathbb{C})$ is a non-constant function.

By Argument principle, it follows that both zeros and poles of $h$ have
multiplicity at least $l+1.$ Since $a\in S_{1},$ we assume, without loss of
generality, that $a=a_{1}.$ Then

$h_{n}(\zeta)=g(z_{n}+\rho_{n}\zeta)=f(z_{n}+\rho_{n}\zeta)-a_{1}\longrightarrow
h(\zeta).$

Case 1. $h$ does not omit zero.

Let $h(\zeta_{0})=0.$ Then we can find some $\delta>0$ such that for
sufficiently large $n,$ $h_{n}$ is holomorphic in $D(\zeta_{0},\delta).$

Let

$\psi_{n}(\zeta)=\frac{h_{n}(\zeta)}{\rho_{n}^{l}}.$

Then $\psi_{n}$ is holomorphic in $D(\zeta_{0},\delta).$ Also,
$\psi_{n}(\zeta^{\prime})=0$ if and only if $h_{n}(\zeta^{\prime})=0$ and so
$h_{n}$ has zeros of multiplicity at least $l+1.$

Next, we claim that $\left\\{\psi_{n}\right\\}$ is not normal at $\zeta_{0}.$
Suppose on the contrary that $\left\\{\psi_{n}\right\\}$ is normal. Then there
exist a $\delta_{1}$ such that $0<\delta_{1}<\delta$ and a subsequence of
$\left\\{\psi_{n}\right\\}$ (again denoted by $\left\\{\psi_{n}\right\\}$)
such that $\psi_{n}\longrightarrow\psi$ locally uniformly in
$D(\zeta_{0},\delta_{1}),$ where $\psi$ is either holomorphic or identically
$\infty$ in $D(\zeta_{0},\delta_{1}).$

Since $h(\zeta_{0})=0$ and $h\not\equiv 0,$ by Hurwitz’s Theorem, there exits
$\zeta_{n}\longrightarrow\zeta_{0}$ such that for sufficiently large $n,$
$h_{n}(\zeta_{n})=0$ and hence

$\psi(\zeta_{0})=\lim\limits_{n\to\infty}\psi_{n}(\zeta_{n})=\lim\limits_{n\to\infty}\frac{h_{n}(\zeta_{n})}{\rho_{n}^{l}}=0.$

Also, since zeros of $h$ in $D(\zeta_{0},\delta_{1})$ are isolated, there is
some $\zeta_{1}\neq\zeta_{0}$ in $D(\zeta_{0},\delta_{1})$ such that
$h(\zeta_{1})\neq 0.$ Thus for sufficiently large $n,$ $\lvert
h_{n}(\zeta_{1})\rvert>0$ and hence

$\lvert\psi_{n}(\zeta_{1})\rvert=\frac{\lvert
h_{n}(\zeta_{1})\rvert}{\rho_{n}^{l}}\longrightarrow\infty.$

This implies that $\left\\{\psi_{n}\right\\}$ converges uniformly to $\infty$
in $D(\zeta_{0},\delta_{1})$ which contradicts the fact that
$\psi(\zeta_{0})=0.$

Now, by Lemma 13, there exists a subsequence of $\left\\{\psi_{n}\right\\}$
(again denoted by $\left\\{\psi_{n}\right\\}$), a sequence of points
$\hat{\zeta_{n}}\longrightarrow\zeta_{0}$ and positive numbers
$r_{n}\longrightarrow 0$ such that

$\phi_{n}(\xi)=\frac{\psi_{n}(\hat{\zeta_{n}}+r_{n}\xi)}{r_{n}^{l}}=\frac{h_{n}(\hat{\zeta_{n}}+r_{n}\xi)}{(\rho_{n}r_{n})^{l}}=\frac{f_{n}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)-a_{1}}{(\rho_{n}r_{n})^{l}}$

converges locally uniformly to a non-constant entire function $\phi(\xi).$ It
is easy to see that $\phi$ has zeros of multiplicity at least $l+1.$ By
Weierstrass’s Theorem, it follows that

$\left(\phi_{n}^{(l)}\right)^{m}(\xi)=\left(f^{(l)}\right)^{m}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi)\longrightarrow\left(\phi^{(l)}\right)^{m}(\xi)$

uniformly on compact subsets of $\mathbb{C}.$

Since $\\#(S_{1})=3,$ by Picard’s Theorem, it follows that
$\left(\phi^{(l)}\right)^{m}$ can omit at most one value from $S_{1}.$ Pick a
non-zero value from $S_{1},$ say $a_{2}$ such that
$\left(\phi^{(l)}\right)^{m}(\xi)=a_{2}$ has a solution.

Let $\xi_{0}\in\mathbb{C}$ be such that
$\left(\phi^{(l)}\right)^{m}(\xi_{0})=a_{2}~{}(\neq 0)$ and
$\phi(\xi_{0})=a_{2}^{*}$ where $a_{2},a_{2}^{*}\in\mathbb{C}.$ Clearly,
$a_{2}^{*}\neq 0,$ otherwise $\left(\phi^{(l)}\right)^{m}(\xi_{0})=0$ owing to
the fact that zeros of $\phi$ have multiplicity at least $l+1.$

Since $\left(\phi^{(l)}\right)^{m}\not\equiv a_{2},$ by Hurwitz’s Theorem,
there exists $\xi_{n}\longrightarrow\xi_{0}$ such that for sufficiently large
$n,$

$\left(f^{(l)}\right)^{m}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=a_{2}~{}(\neq
0).$

Also, $\phi_{n}(\xi_{n})\longrightarrow\phi(\xi_{0})=a_{2}^{*}~{}(\neq 0)$
implies that
$f(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})\longrightarrow a_{1}.$
However, $f(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})\neq a_{1}.$
Indeed, if $f(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=a_{1},$ then
by hypothesis,

$f^{(i)}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=0~{}\mbox{for}~{}1\leq
i\leq l$

and hence

$\left(f^{(l)}\right)^{m}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=0,$

a contradiction to the fact that
$\left(f^{(l)}\right)^{m}(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})=a_{2}\neq
0.$

Thus $\left\\{f(z_{n}+\rho_{n}\hat{\zeta_{n}}+\rho_{n}r_{n}\xi_{n})\right\\}$
is an infinite set and cannot be contained in the finite set $S_{2},$ a
contradiction.

Case 2. $h$ omits zero. Suppose that there exists $\zeta_{0}\in\mathbb{C}$
such that $h(\zeta_{0})=b_{1}-a_{1}.$ Since $h\not\equiv b_{1}-a_{1},$ by
Hurwitz’s Theorem, there exists $\zeta_{n}\longrightarrow\zeta_{0}$ such that
for sufficiently large $n,$ $h_{n}(\zeta_{n})=b_{1}-a_{1}.$ This implies that

$f(z_{n}+\rho_{n}\zeta_{n})=b_{1}\in S_{2}.$

By hypothesis, we have

$\left(f^{(l)}\right)^{m}(z_{n}+\rho_{n}\zeta_{n})\in S_{1}.$

Let

$\left(f^{(l)}\right)^{m}(z_{n}+\rho_{n}\zeta_{n})=a_{3}.$

Then

$\displaystyle\left(h^{(l)}\right)^{m}(\zeta_{0})=\lim\limits_{n\to\infty}\left(h_{n}^{(l)}\right)^{m}(\zeta_{n})$
$\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\rho_{n}^{lm}\left(f^{(l)}\right)^{m}(z_{n}+\rho_{n}\zeta_{n})$
$\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\rho_{n}^{lm}~{}a_{3}=0.$

This shows that

$h(\zeta)=b_{1}-a_{1}\Rightarrow h^{(l)}(\zeta)=0.$

Similarly, it can be shown that

$h(\zeta)=b_{i}-a_{1}\Rightarrow h^{(l)}(\zeta)=0,~{}i=2,3.$

Now applying Lemma 17 to $h,$ we obtain

$\displaystyle 3~{}T(r,h)$ $\displaystyle<$
$\displaystyle\left(l+1\right)\overline{N}(r,h)+\overline{N}\left(r,\frac{1}{h}\right)+N\left(r,\frac{1}{h}\right)+S(r,h)$
$\displaystyle=$ $\displaystyle\overline{N}(r,h)+l\overline{N}(r,h)+S(r,h)$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{l+1}\right)N(r,h)+N(r,h)+S(r,h)$
$\displaystyle\leq$ $\displaystyle\left(\frac{1}{l+1}+1\right)T(r,h)+S(r,h).$

which is a contradiction. Note that the case when $S_{1}=S_{2}$ can occur. If
this is the case then,

$2~{}T(r,h)<\left(\frac{1}{l+1}+1\right)T(r,h)+S(r,h),$

which is again a contradiction. Hence $f$ is a normal function.

$(ii)$ If $a\notin S_{1},$ then by the same argument as in Case 2 of part
$(i)$, one can easily deduce that

$h(\zeta)=a_{i}-a\Rightarrow h^{(l)}(\zeta)=0,~{}i=1,2,3.$

Again, applying Lemma 17 to $h,$ we get

$\displaystyle 3~{}T(r,h)$ $\displaystyle<$
$\displaystyle\left(l+1\right)\overline{N}(r,h)+\overline{N}\left(r,\frac{1}{h}\right)+N\left(r,\frac{1}{h}\right)+S(r,h)$
$\displaystyle\leq$
$\displaystyle\overline{N}(r,h)+l\overline{N}(r,h)+\left(\frac{1}{l+2}\right)N\left(r,\frac{1}{h}\right)+N\left(r,\frac{1}{h}\right)+S(r,h)$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{l+2}\right)N(r,h)+N(r,h)+\left(\frac{1}{l+2}\right)N\left(r,\frac{1}{h}\right)+N\left(r,\frac{1}{h}\right)+S(r,h)$
$\displaystyle\leq$ $\displaystyle\left(\frac{2}{l+2}+2\right)T(r,h)+S(r,h).$

which is a contradiction. Hence $f$ is a normal function.

##  4 Statements and Declaration

Funding: The authors declare that no funds, grants, or other support were
received during the preparation of this manuscript.

Competing Interests: The authors declare that they have no conflict of
interests. The authors have no relevant financial or non-financial interests
to disclose.

Author Contributions: All authors contributed equally to this work.

Data availability: Data sharing not applicable to this article as no datasets
were generated or analysed during the current study.

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