2023

[1]Nikhil Bharti

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

# Value distribution of certain differential polynomials leading to some
normality criteria

Kuldeep Singh Charak<EMAIL_ADDRESS>
Orcid: 0000-0003-0151-3249<EMAIL_ADDRESS>
Orcid: 0000-0003-2501-6247 [

###### Abstract

In this paper, we prove some normality criteria concerning transitivity of
normality from one family of meromorphic functions to another which improve
and generalize some recent results. We also prove some value distribution
results for certain differential polynomials which lead to some normality
criteria involving sharing of holomorphic functions with certain differential
polynomials. As a consequence, a counterexample to the converse of the Bloch’s
principle is also given.

###### keywords:

Meromorphic functions. Normal families. Value distribution theory.
Differential polynomials. Shared functions.

###### pacs:

[

MSC Classification]Primary 30D35. 30D45. 30M04; Secondary 30D30.

##  1 Introduction and Statement of Results

For the sake of convenience, we shall denote by $\mathcal{H}(D),$ the class of
all holomorpic functions on the domain $D$ in $\mathbb{C}$ and by
$\mathcal{M}(D),$ the class of all meromorphic functions on the domain $D.$
$\mathbb{D}$ and $D(a,r)$ shall denote the open unit disk and the open disk
with center $a$ and radius $r,$ in $\mathbb{C}$, respectively. We assume that
the reader is familiar with the standard notations of the Nevanlinna value
distribution theory of meromorphic functions, like
$m(r,f),~{}N(r,f),~{}T(r,f)$ (see hayman ; lo ).

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. The limit
function is either meromorphic in $D$ or identically $\infty.$ If
$\mathcal{F}$ happens to be a family of holomorphic functions, then the
Euclidean metric can be taken in place of the spherical metric, and in this
case, the limit function is either holomorphic in $D$ or identically $\infty$
(see schiff ; zalcman-1 ; zalcman-2 ).

Let $k$ be a positive integer and let $n_{0},n_{1},\ldots,n_{k}$ be non-
negative integers, not all zero. Let $f\in\mathcal{M}(D).$ Then the expression
of the form

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

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

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

are called the degree and weight of the 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 $\gamma_{P}:=\max\left\\{\gamma_{M_{i}}:1\leq i\leq
m\right\\},$ $\nu_{P}:=\min\left\\{\gamma_{M_{i}}:1\leq i\leq m\right\\}$ and
$\Gamma_{P}:=\max\left\\{\Gamma_{M_{i}}:1\leq i\leq m\right\\}$ are the
degree, lower degree and weight of the differential polynomial $P[f],$
respectively. The number $k,$ which is the highest order of derivative
occurring in the differential polynomial $P[f]$ shall be called differential
order of $P[f].$ Further, we shall denote by $\Theta_{P},$ the ratio of the
weight of the differential polynomial to its lower degree. That is,

$\Theta_{P}=\frac{\Gamma_{P}}{\nu_{P}}=\max\left\\{\frac{\Gamma_{M_{i}}}{\gamma_{M_{i}}}:1\leq
i\leq m\right\\}.$

In the present paper, where not otherwise stated, we consider the differential
polynomials of the form

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

where

$M_{i}[f]=\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^{n_{ji}},~{}1\leq i\leq
m~{}$

are normalized differential monomials satisfying

$\frac{\Gamma_{M_{t}}}{\gamma_{M_{t}}}<\frac{\Gamma_{M_{1}}}{\gamma_{M_{1}}}=\Theta_{P}~{}\mbox{
for }~{}2\leq t\leq m,$ (2)

and the coefficients $a_{i}\in\mathcal{H}(D)$ with $a_{1}(z)\neq 0.$

######  Definition 1.

Let $f~{}g\in\mathcal{M}(D).$ Then $f$ and $g$ are said to share the function
$h$ in $D$ if $Z(h,f)=Z(h,g),$ where $Z(h,\psi):=\left\\{\zeta\in
D:\psi(\zeta)-h(\zeta)=0\right\\}$ is the set of zeros of $\psi-h$ in $D$
counted with ignoring multiplicities; if the zeros are counted with their due
multiplicities, then we say that $f$ and $g$ share $h$ with counting
multiplicities and we write this as $f$ and $g$ share $h$ CM. If
$Z(h,f)\subseteq Z(h,g),$ then we say that $f$ and $g$ partially share $h$ in
$D.$

In $2013,$ Liu et al. liu  initiated the study on transitivity of normality
from one family of meromorphic functions to another under suitable conditions
and obtained the following results:

######  Theorem 1.

Let $\mathcal{F},~{}\mathcal{G}\subset\mathcal{M}(D)$ and
$a_{i}~{}(i=1,2,3,4)$ be four distinct complex numbers. Assume that
$\mathcal{G}$ is normal in $D.$ If for every $f\in\mathcal{F},$ there exists
$g\in\mathcal{G},$ such that $f$ and $g$ share $a_{i}~{}(i=1,2,3,4)$ in $D$ ,
then $\mathcal{F}$ is normal in $D.$

######  Theorem 2.

Let $\mathcal{F},~{}\mathcal{G}\subset\mathcal{M}(D)$ be two families, all of
whose zeros have multiplicities at least $k+1,~{}k\in\mathbb{N}.$ Let $a$ be a
non-zero complex number. Assume that the family $\mathcal{G}$ is normal in $D$
such that no subsequence of $\mathcal{G}$ converges spherically locally
uniformly to $\infty$ or to a function $g$ satisfying $g^{(k)}\equiv a.$ If
for every $f\in\mathcal{F},$ there exists $g\in\mathcal{G}$ such that $f$ and
$g$ share $0$ and $\infty,$ and $f^{(k)}$ and $g^{(k)}$ share $a$ CM, then
$\mathcal{F}$ is normal in $D.$

Theorem 1 has been improved where the four distinct shared values are replaced
by four distinct shared holomorphic functions (see (chen-xu, , Theorem 1)) and
also by four uniformly separated proximate values which may depend on each
$f\in\mathcal{F}$ (see (kumar, , Theorem 7)). On the other hand, by a
beautiful application of complex dynamics, Chang (chang-2, , Theorem 1.4)
established that for $k=1,$ the condition “$f^{(k)}$ and $g^{(k)}$ share $a$
CM" in Theorem 2 can be replaced by the condition “$f^{(k)}$ and $g^{(k)}$
share $a$". Following the ideas of Chang chang-2 , Chen and Xu (chen-xu, ,
Theorem 2) improved Theorem 2 as:

######  Theorem 3.

Let $\mathcal{F},~{}\mathcal{G}\subset\mathcal{M}(D)$ be two families, all of
whose zeros have multiplicities at least $k+1,~{}k\in\mathbb{N}.$ Let $a$ be a
non-zero complex number. Assume that the family $\mathcal{G}$ is normal in $D$
such that no subsequence of $\mathcal{G}$ converges spherically locally
uniformly to $\infty$ or to a function $g$ satisfying $g^{(k)}\equiv a,$ and
poles of $f\in\mathcal{F}$ have multiplicities at least $k.$ If for every
$f\in\mathcal{F},$ there exists $g\in\mathcal{G}$ such that $f$ and $g$
partially share $0,$ $f$ and $g$ share $\infty,$ and $f^{(k)}$ and $g^{(k)}$
partially share $a,$ then $\mathcal{F}$ is normal in $D.$

Recently, Ahamed and Mandal (ahamed, , Theorem 2.2) considered families of
holomorphic functions and generalized Theorem 2 to differential monomials as
follows:

######  Theorem 4.

Let $\mathcal{F},~{}\mathcal{G}\subset\mathcal{H}(D)$ be two families, all of
whose zeros have multiplicities at least $k+1,~{}k\in\mathbb{N}.$ Let $a$ be a
non-zero complex number. Assume that the family $\mathcal{G}$ is normal in $D$
such that for any subsequence $g_{n}$ of $\mathcal{G},~{}g_{n}\rightarrow
g,~{}g\not\equiv\infty$ and $M[g]\not\equiv a.$ If for every
$f\in\mathcal{F},$ there exists $g\in\mathcal{G}$ such that $f$ and $g$ share
$0,$ and $M[f]$ and $M[g]$ share $a,$ then $\mathcal{F}$ is normal in $D.$

About Theorem 4, it is natural to ask the following question:

######  Question 1.

Is the family $\mathcal{F}$ normal in $D$ if the differential monomials $M[f]$
and $M[g]$ sharing a non-zero complex number $a$ are replaced by some
differential polynomials partially sharing $a$ in $D$?

We answer Question 1 as follows:

######  Theorem 5.

Let $\mathcal{F},~{}\mathcal{G}\subset\mathcal{H}(D)$ be two families, all of
whose zeros have multiplicities at least $k+1,~{}k\in\mathbb{N}.$ Let $P[f]$
be a differential polynomial defined in (1) and satisfying (2). Let $a$ be a
non-zero complex number. Assume that the family $\mathcal{G}$ is normal in $D$
such that for any subsequence
$\left\\{g_{n}\right\\}\subset\mathcal{G},~{}g_{n}\rightarrow
g,~{}g\not\equiv\infty$ and $P[g]\not\equiv a.$ If for every
$f\in\mathcal{F},$ there exists $g\in\mathcal{G}$ such that $f$ and $g$
partially share $0$, and, $P[f]$ and $P[g]$ partially share $a$ in $D,$ then
$\mathcal{F}$ is normal in $D.$

The following examples show that various conditions in the hypothesis of
Theorem 5 are essential.

######  Example 1.

Let $k$ be a positive integer and $a=1.$ Let

$\mathcal{F}=\left\\{f_{n}:f_{n}(z)=\frac{nz^{k+1}}{(k+1)!},~{}n\in\mathbb{N}\right\\}$

and

$\mathcal{G}=\left\\{g_{n}:g_{n}(z)=\frac{z^{k+1}}{2n(k+1)!},~{}n\in\mathbb{N}\right\\}$

be two families of holomorphic functions in $\mathbb{D}.$ Then for each $n,$
$f_{n}$ and $g_{n}$ have zeros of multiplicities at least $k+1$ and
$g_{n}\rightarrow g\equiv 0.$ Clearly, the family $\mathcal{G}$ is normal in
$\mathbb{D}.$ Let $P[f]:=(k+1)!ff^{(k)}.$ Then

$P[f_{n}](z)=(k+1)!f_{n}f_{n}^{(k)}(z)=n^{2}z^{k+2}\mbox{ and
}P[g_{n}](z)=(k+1)!g_{n}g_{n}^{(k)}(z)=\frac{z^{k+2}}{4n^{2}}.$

One can easily see that $f_{n}(z)=0\Rightarrow g_{n}(z)=0$ and
$P[f_{n}](z)=1\not\Rightarrow P[g_{n}](z)=1.$ However, the family
$\mathcal{F}$ is not normal in $\mathbb{D}.$

This example shows that the condition “$P[f]$ and $P[g]$ partially share $a$
in $D$" in Theorem 5 is essential.

######  Example 2.

Let $k$ be a positive integer and $a=1.$ Let

$\mathcal{F}=\left\\{f_{n}:f_{n}(z)=nz^{k+1},~{}n\in\mathbb{N},~{}n\geq
2\right\\}$

and

$\mathcal{G}=\left\\{g_{n}:g_{n}(z)=\left[z+1-\left(\frac{1}{n^{2}}\right)^{1/(k+2)}\right]^{k+1},~{}n\in\mathbb{N},~{}n\geq
2\right\\}$

be two families of holomorphic functions in $\mathbb{D}.$ Then each $f_{n}$
and $g_{n}$ has zeros of multiplicities at least $k+1$ and $g_{n}\rightarrow
g=(z+1)^{k+1}\not\equiv\infty.$ Obviously, $f_{n}(z)=0\not\Rightarrow
g_{n}(z)=0.$ Let $P[f]:=\frac{1}{(k+1)!}ff^{(k)}.$ Then

$P[f_{n}](z)=n^{2}z^{k+2}=1\Rightarrow
z=\left(\frac{1}{n^{2}}\right)^{1/(k+2)}$

and

$P[g_{n}](z)=\left[z+1-\left(\frac{1}{n^{2}}\right)^{1/(k+2)}\right]^{k+2}=1\mbox{
whenever }z=\left(\frac{1}{n^{2}}\right)^{1/(k+2)}.$

Thus $P[f_{n}](z)=1\Rightarrow P[g_{n}](z)=1.$ Note that the family
$\mathcal{G}$ is normal in $\mathbb{D},$ however, the family $\mathcal{F}$ is
not normal in $\mathbb{D}.$
Thus, the condition “$f$ and $g$ partially share $0$ in $D$" can not be
dropped.

######  Example 3.

Let $k$ be a positive integer and $a$ be any non-zero complex number. Let
$\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ and
$\mathcal{G}=\left\\{g_{n}:n\in\mathbb{N}\right\\}$ be two families of
holomorphic functions in $\mathbb{D},$ where

$f_{n}(z)=\frac{e^{nz}}{n^{k}}\mbox{ and
}g_{n}(z)=\frac{e^{nz}}{n^{k}}+e^{2kn}.$

Then each $f_{n}$ and $g_{n}$ omit zero in $\mathbb{D}$ and hence $f_{n}$ and
$g_{n}$ partially share $0$ in $\mathbb{D}.$ Also, $g_{n}\rightarrow
g\equiv\infty$ and so the family $\mathcal{G}$ is normal in $\mathbb{D}.$ Let
$P[f]:=f^{\prime}+f^{\prime\prime}+\cdots+f^{(k)}.$ Then clearly
$P[f_{n}](z)=P[g_{n}](z)$ and hence $P[f_{n}]$ and $P[g_{n}]$ partially share
$a$ in $\mathbb{D}.$ However, the family $\mathcal{F}$ is not normal in
$\mathbb{D}.$
This shows that the condition “$g\not\equiv\infty$" can not be dropped.

######  Example 4.

Let $k$ be a positive integer and $a$ be a non-zero complex number. Let
$\mathcal{F}=\left\\{f_{n}:f_{n}(z)=e^{nz}/n^{k},~{}n\in\mathbb{N}\right\\}$
and $\mathcal{G}=\left\\{g_{n}:g_{n}(z)=e^{z},~{}n\in\mathbb{N}\right\\}$ be
two families of holomorphic functions in $\mathbb{D}.$ Then $f_{n}$ and
$g_{n}$ omit zero. Clearly, the family $\mathcal{G}$ is normal in $\mathbb{D}$
and $g_{n}\rightarrow g\equiv e^{z}\not\equiv\infty.$ Let
$P[f]:=ae^{-z}f^{(k)}.$ Then $P[g]\equiv a.$ Further,
$P[f_{n}](z)=ae^{-z}f_{n}^{(k)}(z)=ae^{z(n-1)}$ and
$P[g_{n}](z)=ae^{-z}g_{n}^{(k)}(z)=a.$ It follows that $f_{n}(z)=0\Rightarrow
g_{n}(z)=0$ and $P[f_{n}](z)=a\Rightarrow P[g_{n}](z)=a.$ However, the family
$\mathcal{F}$ is not normal in $\mathbb{D}.$
This shows that the condition “$P[g]\not\equiv a$" cannot be removed.

######  Example 5.

Let $m,~{}k$ be two positive integers with $k\geq 2$ and let $a$ be a non-zero
complex number. Let $\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ and
$\mathcal{G}=\left\\{g_{n}:n\in\mathbb{N}\right\\}$ be two families of
holomorphic functions in $\mathbb{D},$ where

$f_{n}(z)=\frac{z^{k+1}}{n(k+1)}+nz\mbox{ and
}g_{n}(z)=\frac{z^{k+1}}{n(k+1)}.$

Then $f_{n}$ has only simple zeros and $g_{n}$ has zeros of multiplicity
$k+1.$ Also, $f_{n}$ and $g_{n}$ partially share $0$ in $\mathbb{D}$ and
$g_{n}\rightarrow g\equiv 0$ and so the family $\mathcal{G}$ is normal in
$\mathbb{D}.$ Let $P[f]:=\left(f^{(k)}\right)^{m}.$ Then clearly
$P[f_{n}](z)=P[g_{n}](z)$ and hence $P[f_{n}]$ and $P[g_{n}]$ partially share
$a$ in $\mathbb{D}.$ However, the family $\mathcal{F}$ is not normal in
$\mathbb{D}.$
Thus the condition“all zeros of each $f\in\mathcal{F}$ have multiplicities at
least $k+1$" cannot be relaxed.

######  Example 6.

Let $k\geq 2$ be a positive integer and $a$ be a non-zero complex number. Let
$\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N},~{}n\geq 2\right\\}$ and
$\mathcal{G}=\left\\{g_{n}:n\in\mathbb{N},~{}n\geq 2\right\\}$ be two families
of holomorphic functions in $\mathbb{D}$ given by

$f_{n}(z)=\frac{nz^{k+1}}{(k+1)!}\mbox{ and
}g_{n}(z)=\frac{z^{k+1}}{(k+1)!}+\frac{az^{k}}{k!}\left(1-\frac{1}{n}\right).$

Then for each $n,$ $f_{n}$ has zeros of multiplicity $k+1$ and $g_{n}$ has
zeros of multiplicity $k.$ Also, $f_{n}$ and $g_{n}$ partially share $0$ in
$\mathbb{D}$ and $g_{n}\rightarrow g\equiv z^{k+1}/(k+1)!+az^{k}/k!$ and so
the family $\mathcal{G}$ is normal in $\mathbb{D}.$ Let $P[f]:=f^{(k)}.$ Then
$P[g](z)=z+a\not\equiv a.$ Now, $P[f_{n}](z)=nz=a\Rightarrow z=a/n$ and
$P[g_{n}](z)=z+a(1-1/n)=a~{}\mbox{whenever }z=a/n.$ Thus $P[f_{n}](z)$ and
$P[g_{n}](z)$ partially share $a$ in $\mathbb{D}.$ However, the family
$\mathcal{F}$ is not normal in $\mathbb{D}.$
This shows that the condition “all zeros of each $g\in\mathcal{G}$ have
multiplicities at least $k+1$" cannot be dropped.

######  Example 7.

Let $a$ be any non-zero complex number. Consider the families
$\mathcal{F}=\left\\{f_{n}:f_{n}(z)=e^{-nz},~{}n\in\mathbb{N}\right\\}$ and
$\mathcal{G}=\left\\{g_{n}:g_{n}(z)=e^{z},~{}n\in\mathbb{N}\right\\}$ of
holomorphic functions in $\mathbb{D}.$ Then for each $n,$ $f_{n}$ and $g_{n}$
omit zero in $\mathbb{D}.$ Also, $g_{n}\rightarrow g\equiv
e^{z}\not\equiv\infty$ and so the family $\mathcal{G}$ is normal in
$\mathbb{D}.$ Let $P[f]:=M_{1}[f]+M_{2}[f],$ where
$M_{1}[f]:=-f^{\prime}f^{(4)}$ and $M_{2}[f]:=f^{\prime\prime}f^{(3)}$ so that
$\Gamma_{M_{1}}/\gamma_{M_{1}}=7/2=\Gamma_{M_{2}}/\gamma_{M_{2}}.$ Then
$P[g]\equiv 0\not\equiv a$ and

$P[f_{n}](z)=\frac{n}{e^{nz}}\cdot\frac{n^{4}}{e^{nz}}+\frac{n^{2}}{e^{nz}}\cdot\frac{-n^{3}}{e^{nz}}=0.$

It follows vacuously that $f_{n}$ and $g_{n}$ partially share $0$ in
$\mathbb{D}$ and $P[f_{n}]$ and $P[g_{n}]$ partially share $a$ in
$\mathbb{D}.$ However, the family $\mathcal{F}$ is not normal in $\mathbb{D}.$
This shows that the condition “$P[f]$ satisfies (2)" cannot be removed.

The meromorphic analogue of Theorem 5 is obtained as:

######  Theorem 6.

Let $\mathcal{F},~{}\mathcal{G}\subset\mathcal{M}(D)$ be two families, all of
whose zeros have multiplicities at least $k+1,~{}k\in\mathbb{N}.$ Let $P[f]$
be a differential polynomial defined in (1) and satisfying (2) such that
$\Theta_{P}>1.$ Let $a$ be a non-zero complex number. Assume that the family
$\mathcal{G}$ is normal in $D$ such that for any subsequence
$\left\\{g_{n}\right\\}\subset\mathcal{G},~{}g_{n}\rightarrow
g,~{}g\not\equiv\infty$ and $P[g]\not\equiv a.$ If for every
$f\in\mathcal{F},$ there exists $g\in\mathcal{G}$ such that $f$ and $g$
partially share $0$ and $\infty$, and $P[f]$ and $P[g]$ parially share $a$ in
$D,$ then $\mathcal{F}$ is normal in $D.$

Examples 1-7 along with the following examples show that all the conditions in
Theorem 6 are essential:

######  Example 8.

Let $k$ be a positive integer and $a=1.$ Let
$\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N},~{}n\geq 2\right\\}$ and
$\mathcal{G}=\left\\{g_{n}:n\in\mathbb{N},~{}n\geq 2\right\\}$ be two families
of meromorphic functions in $\mathbb{D},$ where

$f_{n}(z)=\frac{1}{(-1)^{k}k!~{}nz}\mbox{ and
}g_{n}(z)=\frac{1}{(k+1)!}\left[z+1-\left(\frac{1}{n}\right)^{1/(k+1)}\right]^{k+1}.$

Then for each $n,$ $f_{n}(z)\neq 0$ and $g_{n}$ has zeros of multiplicity
$k+1.$ Obviously, $f_{n}$ and $g_{n}$ partially share $0$ in $\mathbb{D}.$
Also, $g_{n}\rightarrow g=1/(k+1)!(z+1)^{k+1}\not\equiv\infty$ and so the
family $\mathcal{G}$ is normal in $\mathbb{D}.$ Let $P[f]:=f^{(k)}.$ Then
$P[g]=z+1\not\equiv 1.$ Also,

$P[f_{n}](z)=\frac{1}{nz^{k+1}}=1\Rightarrow
z=\left(\frac{1}{n}\right)^{1/(k+1)}$

and

$P[g_{n}](z)=z+1-\left(\frac{1}{n}\right)^{1/(k+1)}=1\mbox{ whenever
}z=\left(\frac{1}{n}\right)^{1/(k+1)}.$

Thus $P[f_{n}]$ and $P[g_{n}]$ partially share $1$ in $\mathbb{D}.$ However,
the family $\mathcal{F}$ is not normal in $\mathbb{D}.$
This example shows that the condition “$f$ and $g$ partially share $\infty$ in
$D$" can not be dropped.

######  Example 9.

Consider the families

$\mathcal{F}=\left\\{f_{n}:f_{n}(z)=\frac{1}{e^{nz}+1},~{}n\in\mathbb{N}\right\\}$

and

$\mathcal{G}=\left\\{g_{n}:g_{n}(z)=\frac{1}{e^{n}\left(e^{nz}+1\right)},~{}n\in\mathbb{N}\right\\}$

of meromorphic functions in $\mathbb{D}.$ Then $f_{n}$ and $g_{n}$ omit zero.
Clearly, $f_{n}$ and $g_{n}$ partially share $\infty$ in $\mathbb{D}.$ Also,
$g_{n}\rightarrow g\equiv 0\not\equiv\infty$ and so the family $\mathcal{G}$
is normal in $\mathbb{D}.$ Let $P[f]:=f.$ Then $\Theta_{P}=1.$ Since $f_{n}$
omits $0$ and $1,$ it follows vacuously that $f_{n}$ and $g_{n}$ partially
share $0$ in $\mathbb{D}$ and $P[f_{n}]$ and $P[g_{n}]$ partially share $1$ in
$\mathbb{D}.$ However, the family $\mathcal{F}$ is not normal in $\mathbb{D}.$
This shows that the condition “$\Theta_{P}>1$" cannot be removed.

######  Remark 1.

The assumption that the coefficients of the differential polynomial $P[f]$
considered in Theorem 5 and Theorem 6 are holomorphic is essential. This can
be seen from the following example:

Consider the families $\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ and
$\mathcal{G}=\left\\{g_{n}:n\in\mathbb{N}\right\\},$ where

$f_{n}(z)=\frac{nz^{k+1}}{(k+1)!}\mbox{ and
}g_{n}(z)=\frac{z^{k+1}}{(k+1)!},~{}z\in\mathbb{D}.$

Then $\mathcal{F},~{}\mathcal{G}\subset\mathcal{H}(\mathbb{D}),$ and $f_{n}$
and $g_{n}$ have zeros of multiplicities $k+1.$ Obviously, the family
$\mathcal{G}$ is normal in $\mathbb{D}$ and $g_{n}\rightarrow g\equiv
z^{k+1}/(k+1)!\not\equiv\infty.$ Let $P[f]:=(1/z)f^{(k)}.$ Then
$P[f_{n}](z)=n\mbox{ and }P[g_{n}](z)=1.$ It follows that $f_{n}$ and $g_{n}$
partially share $0$ and $\infty$ in $\mathbb{D},$ and for any
$a\in\mathbb{C}\setminus\mathbb{N},$ $P[f_{n}]$ and $P[g_{n}]$ partially share
$a$ in $\mathbb{D}.$ However, the family $\mathcal{F}$ is not normal in
$\mathbb{D}.$

Further, the assumption on the coefficients of the differential polynomial
$P[f]$ namely, “$a_{1}(z)\neq 0$" is essential. For example, consider the
families $\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ and
$\mathcal{G}=\left\\{g_{n}:n\in\mathbb{N}\right\\},$ where

$f_{n}(z)=\frac{1}{nz}\mbox{ and }g_{n}(z)=\frac{1}{z},~{}z\in\mathbb{D}.$

Then $\mathcal{F},~{}\mathcal{G}\subset\mathcal{M}(\mathbb{D}),$ and $f_{n}$
and $g_{n}$ omit zero. Obviously, the family $\mathcal{G}$ is normal in
$\mathbb{D}$ and $g_{n}\rightarrow g\equiv 1/z\not\equiv\infty.$ Let
$P[f]:=\frac{z^{k+1}}{(-1)^{k}k!}f^{(k)}.$ Then

$P[f_{n}](z)=\frac{z^{k+1}}{(-1)^{k}k!}\cdot\frac{(-1)^{k}k!}{nz^{k+1}}=\frac{1}{n}\mbox{
and }P[g_{n}](z)=1.$

It follows that $f_{n}$ and $g_{n}$ partially share $0$ and $\infty$ in
$\mathbb{D}$ and for any
$a\in\mathbb{C}\setminus\left\\{1/n:n\in\mathbb{N}\right\\},$ $P[f_{n}]$ and
$P[g_{n}]$ partially share $a$ in $\mathbb{D}.$ However, the family
$\mathcal{F}$ is not normal in $\mathbb{D}.$

In (singh, , Theorem 1.7), Singh and the first author proved the following
theorem:

######  Theorem 7.

Let $n_{0},~{}n_{1},\ldots,~{}n_{k}$ be non-negative integers such that
$n_{0}\geq 2,~{}n_{k}\geq 1$ and $k\in\mathbb{N}.$ Let $h~{}(\not\equiv 0)$ be
a holomorphic function in $D$ with zeros of multiplicity $m~{}(\geq 1).$ Let
$\mathcal{F}\subset\mathcal{M}(D)$ be such that each $f\in\mathcal{F}$ has
zeros of multiplicity at least $k+m+1$ and poles of multiplicity at least
$m+1.$ If for every $f,~{}g\in\mathcal{F},$ $M[f]$ and $M[g]$ share $h$ in
$D,$ then $\mathcal{F}$ is normal in $D.$

We remove the condition on the multiplicities of poles of $f\in\mathcal{F}$
and improve Theorem 7 as:

######  Theorem 8.

Let $n_{0},~{}n_{1},\ldots,~{}n_{k}$ be non-negative integers with $n_{0}\geq
2,~{}n_{k}\geq 1$ and $k\in\mathbb{N}.$ Let $h~{}(\not\equiv 0)$ be a
holomorphic function in $D$ having zeros of multiplicity at most $m~{}(\geq
1).$ Let $\mathcal{F}\subset\mathcal{M}(D)$ be such that zeros of each
$f\in\mathcal{F}$ have multiplicities at least $k+m+1.$ If for every
$f,~{}g\in\mathcal{F},$ $M[f]$ and $M[g]$ share $h$ in $D,$ then $\mathcal{F}$
is normal in $D.$

The inevitability of the condition “$h\not\equiv 0$" in Theorem 8 can be seen
from the following example:

######  Example 10.

Consider the family $\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ of
meromorphic functions on $\mathbb{D}$ given by $f_{n}(z)=e^{nz}.$ Then for
each $n,$ $f_{n}$ omits $0.$ Let $M[f]:=f^{2}f^{\prime}.$ Then
$M[f_{n}](z)=f_{n}^{2}f_{n}^{\prime}=ne^{3nz}.$ Clearly, for distinct
$m,~{}n,$ $M[f_{m}]$ and $M[f_{n}]$ share $h\equiv 0$ in $\mathbb{D}.$
However, the family $\mathcal{F}$ is not normal in $\mathbb{D}.$

The following example demonstrates that the condition “zeros of each
$f\in\mathcal{F}$ have multiplicities at least $k+m+1$" in Theorem 8 cannot be
dropped:

######  Example 11.

Let $k\geq 2$ be a natural number and consider the family
$\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ of meromorphic functions
on $\mathbb{D}$ given by $f_{n}(z)=nz^{k}.$ Then for each $f\in\mathcal{F}$
has zeros of multiplicity $k.$ Let $M[f]:=f^{2}f^{\prime}.$ Then
$M[f_{n}](z)=f_{n}^{2}f_{n}^{\prime}=kn^{3}z^{3k-1}.$ Take $h(z)=z^{3k-1}.$
Then $M[f_{n}](z)-h(z)=z^{3k-1}(kn^{3}-1).$ Clearly, for distinct $m,~{}n,$
$M[f_{m}]$ and $M[f_{n}]$ share $h$ in $\mathbb{D}.$ However, the family
$\mathcal{F}$ is not normal in $\mathbb{D}.$

As an immediate consequence of Theorem 8, we have

######  Corollary 1.

Let $n_{0},~{}n_{1},\ldots,~{}n_{k}$ be non-negative integers with $n_{0}\geq
2,~{}n_{k}\geq 1$ and $k\in\mathbb{N}.$ Let $h$ be non-vanishing holomorphic
function in $D.$ Let $\mathcal{F}\subset\mathcal{M}(D)$ be such that for each
$f\in\mathcal{F},$ $M[f]-h$ has no zero in $D.$ Then $\mathcal{F}$ is normal
in $D.$

It is noteworthy to mention that Corollary 1 leads to a counterexample to the
converse of Bloch’s principle (see bergweiler ; charak-1 ; charak-2 ; charak-3
; lahiri ; li ) which states that if a family
$\mathcal{F}\subset\mathcal{M}(D)$ satisfying a certain property $\mathcal{P}$
in $D$ is normal, then every $f\in\mathcal{M}(\mathbb{C})$ which satisfies
property $\mathcal{P}$ in $\mathbb{C}$ reduces to a constant.

######  Counterexample 1.

Let $D$ be any domain in $\mathbb{C}$ and let
$\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$ be a family of meromorphic
functions in $D$ given by $f_{n}(z)=e^{z}.$ Let $\mathcal{P}$ be the property
that for each $f\in\mathcal{F},$ $M[f]-h$ has no zero in $D,$ where $h$ is a
non-vanishing holomorphic function in $D.$ In view of Corollary 1, the family
$\mathcal{F}$ is normal in $D.$ Now, let $f(z)=e^{z},~{}M[f]=f^{2}f^{\prime}$
and $h(z)=-e^{3z}.$ Then
$f\in\mathcal{H}(\mathbb{C}),~{}h\in\mathcal{H}(\mathbb{C})$ and $h(z)\neq 0.$
Clearly, $M[f](z)-h(z)=2e^{3z}\neq 0,~{}\forall~{}z\in\mathbb{C}.$ Then $f$
satisfies property $\mathcal{P}$ in $\mathbb{C}.$ However, $f$ is non-
constant. This violates the statement of the converse of Bloch’s principle.

##  2 Some Value Distribution Results and Preliminary Lemmas

In this section, we state and prove some results which are crucial to prove
the main results of this paper. Our first preliminary result is an 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 1 (Zalcman-Pang Lemma).

Let $\mathcal{F}\subset\mathcal{M}(D)$ be such that all of its zeros have
multiplicities at least $m$ and all its 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 $z_{n}\rightarrow z_{0}$ and
positive numbers $\rho_{n}$ with $\rho_{n}\rightarrow 0$ such that the
sequence $\left\\{g_{n}\right\\}$ defined by

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

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

The following lemma is due to Hayman (see (hayman-2, , Theorem 3), cf.
(hayman, , Theorem 3.5)) commonly known as “Hayman’s Alternative":

######  Lemma 2.

Let $f\in\mathcal{M}(\mathbb{C})$ be a non-constant function and $k$ be a
positive integer. Then for any $a\in\mathbb{C}\setminus\left\\{0\right\\},$
either $f$ or $f^{(k)}-a$ has at least one zero in $\mathbb{C}.$ Moreover, if
$f$ is transcendental, then $f$ or $f^{(k)}-a$ has infinitely many zeros in
$\mathbb{C}.$

######  Lemma 3.

((wang, , Lemma 6), cf. (bergo, , Theorem 3)) Let
$f\in\mathcal{M}(\mathbb{C})$ be a transcendental function of finite order
having zeros of multiplicity at least $k+1,$ where $k\in\mathbb{N}.$ Then for
$j=1,\ldots,k,$ $f^{(j)}$ assumes every non-zero complex number infinitely
many times.

######  Lemma 4.

(lo, , Lemma 1.2) Let $f_{j}~{}(j=1,~{}2)$ be two non-constant meromorphic
functions in $\mathbb{C}.$ Then

$N(r,f_{1}f_{2})-N\left(r,\frac{1}{f_{1}f_{2}}\right)=N(r,f_{1})+N(r,f_{2})-N\left(r,\frac{1}{f_{1}}\right)-N\left(r,\frac{1}{f_{2}}\right).$

######  Lemma 5.

(singh, , Lemma 2.1) Let $n_{0},n_{1},\ldots,n_{k},m$ be non-negative integers
with $n_{0}\geq 2,~{}\sum\limits_{j=1}^{k}n_{j}\geq 1$ and $k\in\mathbb{N}.$
Let $p~{}(\not\equiv 0)$ be a polynomial of degree $m$ and $f$ be a non-
constant rational function with zeros of multiplicity at least $k+m$ and poles
of multiplicity at least $m+1.$ Let
$M[f]=\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^{n_{j}}$ be a differential
monomial of $f.$ Then $M[f]-p$ has at least two distinct zeros in
$\mathbb{C}.$

######  Lemma 6.

Let $f$ be a non-constant and non-vanishing rational function in $\mathbb{C}$
and let $p~{}(\not\equiv 0)$ be a polynomial. Let
$M[f]=\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^{n_{j}}$ be a differential
monomial of $f,$ where $k\in\mathbb{N}$ and $n_{j}$’s ($j=0,1,\ldots,k$) are
non-negative integers such that $\sum\limits_{j=0}^{k}n_{j}\geq 1.$ Then
$M[f]-p$ has at least $\Gamma_{M}$ distinct zeros in $\mathbb{C}.$

Proof: The proof is based on the method of Chang chang  (see also deng ) with
significant modifications. Since the computations involved in the proof are
intricate, we describe the proof in detail.

Since $f$ is non-vanishing, $f$ cannot be a polynomial and so $f$ has at least
one pole. Thus we can write

$f(z)=\frac{A}{\prod_{i=1}^{t}(z+z_{i})^{t_{i}}}$ (3)

and

$p(z)=B\prod\limits_{i=1}^{d}(z+v_{i})^{d_{i}},$ (4)

where $A,~{}B$ are non-zero constants, $d,~{}t,~{}t_{i}$ are positive integers
and $d_{i}$ are non-negative integers. Also, $v_{i}$ (when $1\leq i\leq d$)
are distinct complex numbers and $z_{i}$ (when $1\leq i\leq t$) are distinct
complex numbers.

From (3), we deduce that

$M[f](z)=\frac{g(z)}{\prod_{i=1}^{t}(z+z_{i})^{(t_{i}-1)\gamma_{M}+\Gamma_{M}}},$
(5)

where $g$ is a polynomial of degree $(t-1)(\Gamma_{M}-\gamma_{M}).$

Also, by assumption, it easily follows that $M[f]-p$ has at least one zero in
$\mathbb{C}.$ Thus we may set

$M[f](z)=p(z)+\frac{C\prod\limits_{i=1}^{s}(z+w_{i})^{l_{i}}}{\prod_{i=1}^{t}(z+z_{i})^{(t_{i}-1)\gamma_{M}+\Gamma_{M}}},$
(6)

where $C(\neq 0)\in\mathbb{C},$ $l_{i}$ are positive integers and $w_{i}$
($1\leq i\leq s$) are distinct complex numbers.

Let $D=\sum\limits_{i=1}^{d}d_{i}$ and $T=\sum\limits_{i=1}^{t}t_{i}.$ Then
from (4), (5) and (6), we get

$B\prod\limits_{i=1}^{d}(z+v_{i})^{d_{i}}\prod_{i=1}^{t}(z+z_{i})^{(t_{i}-1)\gamma_{M}+\Gamma_{M}}+C\prod\limits_{i=1}^{s}(z+w_{i})^{l_{i}}=g(z)$
(7)

By simple calculation, one can easily see that

$\sum\limits_{i=1}^{s}l_{i}=\sum\limits_{i=1}^{t}\left[(t_{i}-1)\gamma_{M}+\Gamma_{M}\right]+\sum\limits_{i=1}^{d}d_{i}=(T-t)\gamma_{M}+t\Gamma_{M}+D$

and $C=-B.$

Also, from (7), we get

$\prod\limits_{i=1}^{d}(1+v_{i}r)^{d_{i}}\prod_{i=1}^{t}(1+z_{i}r)^{(t_{i}-1)\gamma_{M}+\Gamma_{M}}-\prod\limits_{i=1}^{s}(1+w_{i}r)^{l_{i}}=r^{\Gamma_{M}+(T-1)\gamma_{M}+D}h(r),$

where $h(r):=r^{(t-1)(\Gamma_{M}-\gamma_{M})}g(1/r)/B$ is a polynomial of
degree less than $(t-1)(\Gamma_{M}-\gamma_{M}).$ Furthermore, it follows that

$\displaystyle\frac{\prod\limits_{i=1}^{d}(1+v_{i}r)^{d_{i}}\prod\limits_{i=1}^{t}(1+z_{i}r)^{(t_{i}-1)\gamma_{M}+\Gamma_{M}}}{\prod\limits_{i=1}^{s}(1+w_{i}r)^{l_{i}}}$
$\displaystyle=1+\frac{r^{\Gamma_{M}+(T-1)\gamma_{M}+D}h(r)}{\prod\limits_{i=1}^{s}(1+w_{i}r)^{l_{i}}}$
$\displaystyle=1+O\left(r^{\Gamma_{M}+(T-1)\gamma_{M}+D}\right)$ (8)

as $r\rightarrow 0.$ Taking logarithmic derivatives of both sides of (2), we
obtain

$\sum\limits_{i=1}^{d}\frac{d_{i}v_{i}}{1+v_{i}r}+\sum\limits_{i=1}^{t}\frac{\left[(t_{i}-1)\gamma_{M}+\Gamma_{M}\right]z_{i}}{1+z_{i}r}-\sum\limits_{i=1}^{s}\frac{l_{i}w_{i}}{1+w_{i}r}=O\left(r^{\Gamma_{M}+(T-1)\gamma_{M}+D-1}\right)$
(9)

as $r\rightarrow 0.$

Let
$S_{1}=\left\\{v_{1},v_{2},\ldots,v_{d}\right\\}\cap\left\\{z_{1},z_{2},\ldots,z_{t}\right\\}$
and
$S_{2}=\left\\{v_{1},~{}v_{2},\ldots,v_{d}\right\\}\cap\left\\{w_{1},w_{2},\ldots,w_{s}\right\\}.$
Then the following four cases arise:

Case 1: $S_{1}=S_{2}=\phi$. Let $z_{t+i}=v_{i}$ when $1\leq i\leq d$ and

$T_{i}=\left\\{\begin{array}[]{cc}(t_{i}-1)\gamma_{M}+\Gamma_{M}&\mbox{if}~{}1\leq
i\leq t,\\\ d_{i-t}&\mbox{if}~{}t+1\leq i\leq t+d.\end{array}\right.$

Then (9) can be written as

$\sum\limits_{i=1}^{t+d}\frac{T_{i}z_{i}}{1+z_{i}r}-\sum\limits_{i=1}^{s}\frac{l_{i}w_{i}}{1+w_{i}r}=O\left(r^{\Gamma_{M}+(T-1)\gamma_{M}+D-1}\right)~{}\mbox{as
$r\rightarrow 0.$}$ (10)

Comparing the coefficients of
$r^{j},~{}j=0,1,\ldots,\Gamma_{M}+(T-1)\gamma_{M}+D-2$ in (10), we find that

$\sum\limits_{i=1}^{t+d}T_{i}z^{j}-\sum\limits_{i=1}^{s}l_{i}w^{j}=0,~{}\mbox{for
each}~{}j=1,2,\ldots,\Gamma_{M}+(T-1)\gamma_{M}+D-1.$ (11)

Let $z_{t+d+i}=w_{i},~{}1\leq i\leq s.$ Then from (11), we deduce that the
system of linear equations

$\sum\limits_{i=1}^{t+d+s}z_{i}^{j}x_{i}=0,~{}j=0,1,\ldots,\Gamma_{M}+(T-1)\gamma_{M}+D-1,$
(12)

has a non-zero solution

$\left(x_{1},\ldots,x_{t+d},x_{t+d+1},\ldots,x_{t+d+s}\right)=\left(T_{1},\ldots,T_{t+d},-l_{1},\ldots,-l_{s}\right).$

If $\Gamma_{M}+(T-1)\gamma_{M}+D\geq t+d+s,$ then by Cramer’s rule (see
(mirsky, , p. 134)), for $0\leq j\leq\Gamma_{M}+(T-1)\gamma_{M}+D-1,$ the
determinant det$(z_{i}^{j})_{(t+d+s)\times(t+d+s)}$ of the coefficients of the
system (12) is equal to zero. However, since $z_{i},~{}1\leq i\leq t+d+s$ are
distinct and det$(z_{i}^{j})_{(t+d+s)\times(t+d+s)}$ is a Vandermonde
determinant (see (mirsky, , p. 17)), so it cannot be equal to zero, a
contradiction. Hence $\Gamma_{M}+(T-1)\gamma_{M}+D<t+d+s.$ Since
$T=\sum\limits_{i=1}^{t}t_{i}\geq t$ and $D=\sum\limits_{i=1}^{d}d_{i}\geq d,$
it follows that $s\geq\Gamma_{M}.$

Case 2: $S_{1}\neq\phi$ and $S_{2}=\phi.$ We may assume, without loss of
generality, that $S_{1}=\left\\{v_{1},v_{2},\ldots,v_{m_{1}}\right\\}.$ Then
$v_{i}=z_{i}$ for $1\leq i\leq m_{1}.$ Take $m_{3}=d-m_{1}.$

Subcase 2.1: $m_{3}\geq 1.$ Let $z_{t+i}=v_{m_{1}+i}$ for $1\leq i\leq m_{3}$
and if $m_{1}<t,$ then set

$T_{i}=\left\\{\begin{array}[]{ccc}(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i}&\mbox{if
}1\leq i\leq m_{1},\\\ (t_{i}-1)\gamma_{M}+\Gamma_{M}&\mbox{if }m_{1}+1\leq
i\leq t,\\\ d_{m_{1}-t+i}&\mbox{if }t+1\leq i\leq t+m_{3}.\end{array}\right.$

If $m_{1}=t,$ then set

$T_{i}=\left\\{\begin{array}[]{cc}(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i}&\mbox{if
}1\leq i\leq m_{1}=t,\\\ d_{m_{1}-t+i}&\mbox{if }t+1\leq i\leq
t+m_{3}.\end{array}\right.$

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

$T_{i}=\left\\{\begin{array}[]{cc}(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i}&\mbox{if
}1\leq i\leq m_{1},\\\ (t_{i}-1)\gamma_{M}+\Gamma_{M}&\mbox{if }m_{1}+1\leq
i\leq t\end{array}\right.$

and if $m_{1}=t,$ then set
$T_{i}=(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i},~{}\mbox{for }1\leq i\leq
m_{1}=t.$

Thus (9) can be written as:

$\sum\limits_{i=1}^{t+m_{3}}\frac{T_{i}z_{i}}{1+z_{i}r}-\sum\limits_{i=1}^{s}\frac{l_{i}w_{i}}{1+w_{i}r}=O\left(r^{\Gamma_{M}+(T-1)\gamma_{M}+D-1}\right)~{}\mbox{as
$r\rightarrow 0,$}$

where $0\leq m_{3}\leq d-1.$ Proceeding in similar fashion as in Case 1, we
deduce that $s\geq\Gamma_{M}.$

Case 3: $S_{1}=\phi$ and $S_{2}\neq\phi.$ We may assume, without loss of
generality, that $S_{2}=\left\\{v_{1},v_{2},\ldots,v_{m_{2}}\right\\}.$ Then
$v_{i}=w_{i}$ for $1\leq i\leq m_{2}.$ Take $m_{4}=d-m_{2}.$

Subcase 3.1: $m_{4}\geq 1.$ Let $w_{s+i}=v_{m_{2}+i}$ for $1\leq i\leq m_{4}$
and if $m_{2}<s,$ then set

$L_{i}=\left\\{\begin{array}[]{ccc}l_{i}-d_{i}&\mbox{if }1\leq i\leq m_{2},\\\
l_{i}&\mbox{if }m_{2}+1\leq i\leq s,\\\ -d_{m_{2}-s+i}&\mbox{if }s+1\leq i\leq
s+m_{4}.\end{array}\right.$

If $m_{2}=s,$ then set

$L_{i}=\left\\{\begin{array}[]{cc}l_{i}-d_{i}&\mbox{if }1\leq i\leq
m_{2}=s,\\\ -d_{m_{2}-s+i}&\mbox{if }s+1\leq i\leq s+m_{4}.\end{array}\right.$

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

$L_{i}=\left\\{\begin{array}[]{cc}l_{i}-d_{i}&\mbox{if }1\leq i\leq m_{2},\\\
l_{i}&\mbox{if }m_{2}+1\leq i\leq s\end{array}\right.$

and if $m_{2}=s,$ then set $L_{i}=l_{i}-d_{i},~{}\mbox{for }1\leq i\leq
m_{2}=s.$

Thus (9) can be written as:

$\sum\limits_{i=1}^{t}\frac{\left[(t_{i}-1)\gamma_{M}+\Gamma_{M}\right]z_{i}}{1+z_{i}r}-\sum\limits_{i=1}^{s+m_{4}}\frac{L_{i}w_{i}}{1+w_{i}r}=O\left(r^{\Gamma_{M}+(T-1)\gamma_{M}+D-1}\right)~{}\mbox{as
$r\rightarrow 0,$}$

where $0\leq m_{4}\leq d-1.$ Proceeding in similar way as in Case 1, we deduce
that $s\geq\Gamma_{M}.$

Case 4. $S_{1}\neq\phi$ and $S_{2}\neq\phi.$ We may assume, without loss of
generality, that
$S_{1}=\left\\{v_{1},v_{2},\ldots,v_{m_{1}}\right\\},~{}S_{2}=\left\\{v_{1},v_{2},\ldots,v_{m_{2}}\right\\}.$
Then $v_{i}=z_{i}$ for $1\leq i\leq m_{1}$ $w_{i}=v_{m_{1}+i}$ for $1\leq
i\leq m_{2}.$ Take $m_{5}=d-m_{2}-m_{1}.$

Subcase 4.1: $m_{5}\geq 1.$ Let $z_{t+i}=v_{m_{1}+m_{2}+i}$ for $1\leq i\leq
m_{5}$ and if $m_{1}<t,$ then set

$T_{i}=\left\\{\begin{array}[]{ccc}(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i}&\mbox{if
}1\leq i\leq m_{1},\\\ (t_{i}-1)\gamma_{M}+\Gamma_{M}&\mbox{if }m_{1}+1\leq
i\leq t,\\\ d_{m_{1}+m_{2}-t+i}&\mbox{if }t+1\leq i\leq
t+m_{5}.\end{array}\right.$

If $m_{1}=t,$ then set

$T_{i}=\left\\{\begin{array}[]{cc}(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i}&\mbox{if
}1\leq i\leq m_{1}=t,\\\ d_{m_{1}+m_{2}-t+i}&\mbox{if }t+1\leq i\leq
t+m_{5}.\end{array}\right.$

If $m_{2}<s,$ then set

$L_{i}=\left\\{\begin{array}[]{cc}l_{i}-d_{m_{1}+i}&\mbox{if }1\leq i\leq
m_{2},\\\ l_{i}&\mbox{if }m_{2}+1\leq i\leq s\end{array}\right.$

and if $m_{2}=s,$ then set $L_{i}=l_{i}-d_{m_{1}+i},~{}\mbox{for }1\leq i\leq
m_{2}=s.$

Subcase 4.2: $m_{5}=0.$ If $m_{1}<t,$ then set

$T_{i}=\left\\{\begin{array}[]{cc}(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i}&\mbox{if
}1\leq i\leq m_{1},\\\ (t_{i}-1)\gamma_{M}+\Gamma_{M}&\mbox{if }m_{1}+1\leq
i\leq t.\end{array}\right.$

If $m_{1}=t,$ then set $T_{i}=(t_{i}-1)\gamma_{M}+\Gamma_{M}+d_{i}~{}\mbox{for
}1\leq i\leq m_{1}=t.$

Also, if $m_{2}<s,$ then set

$L_{i}=\left\\{\begin{array}[]{cc}l_{i}-d_{m_{1}+i}&\mbox{if }1\leq i\leq
m_{2},\\\ l_{i}&\mbox{if }m_{2}+1\leq i\leq s\end{array}\right.$

and if $m_{2}=s,$ then set $L_{i}=l_{i}-d_{m_{1}+i},~{}\mbox{for }1\leq i\leq
m_{2}=s.$

Then in both subcases, (9) can be written as

$\sum\limits_{i=1}^{t+m_{5}}\frac{T_{i}z_{i}}{1+z_{i}r}-\sum\limits_{i=1}^{s}\frac{L_{i}w_{i}}{1+w_{i}r}=O\left(r^{\Gamma_{M}+(T-1)\gamma_{M}+D-1}\right)~{}\mbox{as
$r\rightarrow 0,$}$

where $0\leq m_{5}\leq d-2.$ Proceeding in similar fashion as in Case 1, we
deduce that $s\geq\Gamma_{M}.$ This completes the proof.

The following lemma is an extension of a result of Deng et al (deng-2, , Lemma
5) to a more general differential monomial:

######  Lemma 7.

Let $k$ be a positive integer and $n_{0},~{}n_{1},\ldots,~{}n_{k}$ be non-
negative integers with $n_{0}\geq 2,~{}n_{k}\geq 1.$ Let $f$ be a non-constant
meromorphic function in $\mathbb{C}$ and $p$ be a polynomial of degree $m\geq
1.$ Let $M[f]=\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^{n_{j}}$ be a
differential monomial of $f.$ If $f$ has zeros of multiplicity at least $k+m,$
then $M[f]-p$ has at least two distinct zeros in $\mathbb{C}.$

Proof: Since $f\not\equiv 0,$ we can write

$\frac{1}{f^{\gamma_{M}}}=\frac{M[f]}{pf^{\gamma_{M}}}-\frac{pM[f]^{\prime}-p^{\prime}M[f]}{p~{}f^{\gamma_{M}}}\cdot\frac{M[f]-p}{pM[f]^{\prime}-p^{\prime}M[f]},$

where $\gamma_{M}=n_{0}+n_{1}+\cdots+n_{k}.$

Then

$\displaystyle\gamma_{M}~{}m\left(r,\frac{1}{f}\right)$ $\displaystyle\leq
m\left(r,\frac{M[f]}{pf^{\gamma_{M}}}\right)+m\left(r,\frac{p~{}M[f]^{\prime}-p^{\prime}M[f]}{pf^{\gamma_{M}}}\right)$
$\displaystyle\qquad+m\left(r,\frac{M[f]-p}{p~{}M[f]^{\prime}-p^{\prime}M[f]}\right)+O(1).$

From the Nevanlinna’s theorem on logarithmic derivative, we have
$m\left(r,\frac{M[f]}{f^{\gamma_{M}}}\right)=S(r,f).$ Also, using
$m(r,p)=m\log{r}+O(1),~{}m\left(r,\frac{1}{p}\right)=O(1)$ and Lemma 4, we get

$\displaystyle\gamma_{M}~{}m\left(r,\frac{1}{f}\right)$ $\displaystyle\leq
m\left(r,\frac{M[f]-p}{p~{}M[f]^{\prime}-p^{\prime}M[f]}\right)+S(r,f)$
$\displaystyle=m\left(r,\frac{p~{}M[f]^{\prime}-p^{\prime}M[f]}{M[f]-p}\right)+N\left(r,\frac{p~{}M[f]^{\prime}-p^{\prime}M[f]}{M[f]-p}\right)$
$\displaystyle\qquad-~{}N\left(r,\frac{M[f]-p}{p~{}M[f]^{\prime}-p^{\prime}M[f]}\right)+S(r,f)$
$\displaystyle=m\left(r,\frac{p\left[\frac{M[f]}{p}-1\right]^{\prime}}{\left[\frac{M[f]}{p}-1\right]}\right)+N\left(r,p~{}M[f]^{\prime}-p^{\prime}M[f]\right)+N\left(r.\frac{1}{M[f]-p}\right)$
$\displaystyle\qquad-~{}N\left(r,\frac{1}{p~{}M[f]^{\prime}-p^{\prime}M[f]}\right)-N\left(r,M[f]-p\right)+S(r,f)$
$\displaystyle\leq\overline{N}(r,f)+N\left(r,\frac{1}{M[f]-p}\right)-N\left(r,\frac{1}{p~{}M[f]^{\prime}-p^{\prime}M[f]}\right)$
$\displaystyle\qquad+m\log{r}+S(r,f).$

This gives

$\displaystyle\gamma_{M}~{}T(r,f)$
$\displaystyle\leq\overline{N}(r,f)+\gamma_{M}~{}N\left(r,\frac{1}{f}\right)+N\left(r,\frac{1}{M[f]-p}\right)$
$\displaystyle\qquad-N\left(r,\frac{1}{pM[f]^{\prime}-p^{\prime}M[f]}\right)+m\log{r}+S(r,f).$
(13)

Note that if $z_{0}$ is a zero of $f$ with multiplicity $l$ ($\geq k+m$), then
$z_{0}$ is a zero of $pM[f]^{\prime}-p^{\prime}M[f]$ with multiplicity at
least $\gamma_{M}l-(\Gamma_{M}-\gamma_{M})-1,$ where
$\Gamma_{M}-\gamma_{M}=n_{1}+2n_{2}+\cdots+kn_{k}.$ Therefore,

$N\left(r,\frac{1}{p~{}M[f]^{\prime}-p^{\prime}M[f]}\right)\geq\left[\gamma_{M}l-(\Gamma_{M}-\gamma_{M})-1\right]\overline{N}\left(r,\frac{1}{f}\right).$

Also, since
$pM[f]^{\prime}-p^{\prime}M[f]=p\left(M[f]-p\right)^{\prime}-p^{\prime}\left(M[f]-p\right),$
we have

$N\left(r,\frac{1}{p~{}M[f]^{\prime}-p^{\prime}M[f]}\right)\geq
N\left(r,\frac{1}{M[f]-p}\right)-\overline{N}\left(r,\frac{1}{M[f]-p}\right).$

Thus from (2), we obtain

$\displaystyle\gamma_{M}~{}T(r,f)$
$\displaystyle\leq\overline{N}(r,f)+(\Gamma_{M}-\gamma_{M}+1)\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{M[f]-p]}\right)$
$\displaystyle\qquad+~{}m\log{r}+S(r,f)$
$\displaystyle\leq\overline{N}(r,f)+(\Gamma_{M}-\gamma_{M}+1)\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{M[f]-p]}\right)$
$\displaystyle\qquad+m\log{r}+S(r,f)$
$\displaystyle\leq\overline{N}(r,f)+\frac{\Gamma_{M}-\gamma_{M}+1}{k+m}~{}N\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{M[f]-p]}\right)$
$\displaystyle\qquad+m\log{r}+S(r,f).$ (14)

Thus

$\left[(\gamma_{M}-1)-\frac{\Gamma_{M}-\gamma_{M}+1}{k+m}\right]T(r,f)\leq\overline{N}\left(r,\frac{1}{M[f]-p]}\right)+m\log{r}+S(r,f).$

That is,

$C~{}T(r,f)\leq\overline{N}\left(r,\frac{1}{M[f]-p]}\right)+m\log{r}+S(r,f),$
(15)

where

$\displaystyle C$
$\displaystyle:=(\gamma_{M}-1)-\frac{\Gamma_{M}-\gamma_{M}+1}{k+m}$
$\displaystyle=(n_{0}+n_{1}+n_{2}+\cdots+n_{k}-1)-\frac{(n_{1}+2n_{2}+\cdots+kn_{k})+1}{k+m}$
$\displaystyle=n_{0}-1+\frac{(k+m-1)n_{1}+(k+m-2)n_{2}+\cdots+(m+1)n_{k-1}+mn_{k}-1}{k+m}$
$\displaystyle=n_{0}-1+\frac{\alpha+mn_{k}-1}{k+m},$

wherein $\alpha:=(k+m-1)n_{1}+(k+m-2)n_{2}+\cdots+(m+1)n_{k-1}\geq 0.$

Now suppose that $M[f]-p$ has at most one distinct zero. Then we consider the
following cases:

Case 1: $\alpha\neq 0.$ Then

$C\geq 1+\frac{\alpha}{k+m}$

and so from (15), we obtain

$T(r,f)<C~{}T(r,f)\leq(m+1)\log{r}+S(r,f).$

This implies that $f$ is a rational function of degree at most $m.$ However,
zeros of $f$ have multiplicities at least $k+m\geq m+1.$ Hence $f$ must be
non-vanishing and so by Lemma 6, $M[f]-p$ has at least
$n_{0}+2n_{1}+\cdots+(k+1)n_{k}\geq 4$ distinct zeros, a contradiction.

Case 2: $\alpha=0.$ Then

$C=n_{0}-1+\frac{mn_{k}-1}{k+m}.$

If either $n_{k}\geq 2$ or $m\geq 2,$ then

$C\geq 1+\frac{1}{k+m}$

and so by the same argument as in Case 1, we get a contradiction. Thus we only
need to consider the case when $n_{k}=1$ and $m=1.$ So, let $n_{k}=m=1.$ Then
from (15), we obtain

$(n_{0}-1)T(r,f)\leq\overline{N}\left(r,\frac{1}{M[f]-p]}\right)+\log{r}+S(r,f).$

If $M[f](z)-p(z)\neq 0,$ then

$T(r,f)\leq(n_{0}-1)T(r,f)\leq\log{r}+S(r,f),$

showing that $f$ is a rational function of degree at most one. Since zeros of
$f$ have multiplicities at least $k+1\geq 2,$ it follows that $f$ must be non-
vanishing and so by Lemma 6, $M[f]-p$ has at least
$n_{0}+2n_{1}+\cdots+(k+1)n_{k}\geq 4$ distinct zeros, a contradiction. Thus
by the assumption, $M[f]-p$ has exactly one zero. Then from (2), we have

$n_{0}T(r,f)\leq\overline{N}(r,f)+2\log{r}+S(r,f).$ (16)

Subcase 2.1: $n_{0}\geq 3.$ Then from (16), we have

$T(r,f)\leq\log{r}+S(r,f),$

showing that $f$ is a rational function of degree at most one. Again, since
zeros of $f$ have multiplicities at least $k+1\geq 2,$ $f$ must be non-
vanishing and so by Lemma 6, $M[f]-p$ has at least
$n_{0}+2n_{1}+\cdots+(k+1)n_{k}\geq 5$ distinct zeros, a contradiction.

Subcase 2.2: $n_{0}=2.$ Then (16) gives

$T(r,f)\leq 2\log{r}+S(r,f).$

This means that $f$ is a rational function of degree at most $2.$ If $k\geq
2,$ then the fact that $f$ has zeros of multiplicity at least $k+1\geq 3$
implies that $f$ is non-vanishing and hence by Lemma 6, $M[f]-p$ has at least
$n_{0}+2n_{1}+\cdots+(k+1)n_{k}\geq 5$ distinct zeros, a contradiction. Hence
$k=1.$

Now since $f$ is a rational function of degree at most $2,$ it follows that
either $f$ is non-vanishing or $f$ has only one distinct zero with
multiplicity $2.$ If $f(z)\neq 0,$ then by Lemma 6, $M[f]-p$ has at least
$n_{0}+2n_{1}+\cdots+(k+1)n_{k}\geq 4$ distinct zeros, a contradiction. Hence
$f$ has the following forms:

$A_{1}:f(z)=a(z-z_{0})^{2},~{}~{}A_{2}:f(z)=\frac{a(z-z_{0})^{2}}{z-z_{1}}\mathchar
24635\relax\;$
$A_{3}:f(z)=\frac{a(z-z_{0})^{2}}{(z-z_{1})^{2}},~{}~{}A_{4}:f(z)=\frac{a(z-z_{0})^{2}}{(z-z_{1})(z-z_{2})},$

where $a~{}(\neq 0)\in\mathbb{C}.$ Clearly, $T(r,f)=2\log{r}+O(1).$

If $f$ is of the form $A_{1},$ then $\overline{N}\left(r,f\right)=0$ and
$\overline{N}\left(r,\frac{1}{f}\right)\leq 1/2~{}T(r,f)+O(1).$ Also, from
(2), it follows that

$3~{}T(r,f)\leq\overline{N}(r,f)+2~{}\overline{N}\left(r,\frac{1}{f}\right)+2\log{r}+S(r,f).$

This then implies that $T(r,f)\leq\log{r}+S(r,f),$ a contradiction. Similarly,
if $f$ is of the form $A_{2}$ or $A_{3}$, then we obtain $T(r,f)\leq
4/3\log{r}+S(r,f),$ which is again a contradiction.

Hence

$f(z)=\frac{a(z-z_{0})^{2}}{(z-z_{1})(z-z_{2})}.$

Then

$M[f](z)=f^{2}(z)f^{\prime}(z)=\frac{a^{3}(z-z_{0})^{5}[(2z_{0}-z_{1}-z_{2})z+2z_{1}z_{2}-z_{0}(z_{1}+z_{2})]}{(z-z_{1})^{4}(z-z_{2})^{4}}$
(17)

Since $p$ is a polynomial of degree $m=1,$ we set $p(z)=b(z-z_{3}),$ where
$b~{}(\neq 0),~{}z_{3}\in\mathbb{C}.$ Also, since $M[f]-p$ has exactly one
zero, we can write

$M[f](z)=b(z-z_{3})+\frac{d(z-z_{4})^{r}}{(z-z_{1})^{4}(z-z_{2})^{4}}$ (18)

By simple calculation, one can easily see that $d=-b,~{}r=9$ and $z_{4}\neq
z_{0}.$ Then from (17), we obtain

$M[f]^{\prime\prime}(z)=\frac{(z-z_{0})^{3}g_{1}(z)}{(z-z_{1})^{6}(z-z_{2})^{6}},$
(19)

where $g_{1}$ is a polynomial of degree at most $5.$

Also, from (18), we get

$M[f]^{\prime\prime}(z)=\frac{(z-z_{4})^{7}g_{2}(z)}{(z-z_{1})^{6}(z-z_{2})^{6}},$
(20)

where $g_{2}$ is a polynomial of degree at most $4.$

Comparing (19) and (20) and using the fact that $z_{4}\neq z_{0},$ we find
that $g_{1}$ is a polynomial of degree at least $7,$ a contradiction. This
completes the proof.

The following lemma is a generalization of Hayman’s Alternative and Lemma 3 to
differential polynomials:

######  Lemma 8.

Let $f\in\mathcal{M}(\mathbb{C})$ be a transcendental function and
$\psi~{}(\not\equiv 0,\infty)$ be a small function of $f$ (that is,
$T(r,\psi)=S(r,f)$ as $r\rightarrow\infty$). Let $P[f]$ be a differential
polynomial of $f$ having differential order $k$ and with meromorphic
coefficients $b_{j}$ as small functions of $f.$ Assume that $P[f]$ is non-
constant. Then the following conclusions hold:

  * (i)

If $\nu_{P}>1,$ then either $f$ or $P[f]-\psi$ has infinitely many zeros in
$\mathbb{C}.$ Moreover, if $f$ is entire, then we can take $\nu_{P}\geq 1.$

  * (ii)

If $f$ has zeros of multiplicity at least $k+1,$ and
$(k+2)\nu_{P}>\Gamma_{P}+2,$ then $P[f]-\psi$ has infinitely many zeros in
$\mathbb{C}.$ If $f$ is entire, then the condition $(k+2)\nu_{P}>\Gamma_{P}+2$
can be replaced by the condition $(k+2)\nu_{P}>\Gamma_{P}+1.$

The conditions $\nu_{p}>1$ in (i) and $(k+2)\nu_{P}>\Gamma_{P}+2$ in (ii) are
essential. For example, let $f(z)=e^{z}/(1+e^{z}).$ Then $f(z)\neq 0,~{}1.$
Let $P[f]:=2f-f^{2}$ so that $\nu_{p}=1$ and $(k+2)\nu_{P}<\Gamma_{P}+2.$
However, $P[f](z)=(2e^{z}+e^{2z})/(1+e^{z})^{2}\neq 1.$

Proof: [Proof of Lemma 8] Write $P[f]$ in the form

$P[f]=H_{\nu_{P}}[f]+H_{\nu_{P}+1}[f]+\cdots+H_{\gamma_{P}}[f],$

where $H_{j}[f]$’s are homogeneous differential polynomials of $f$ of degree
$j.$

Since $f\not\equiv 0,$ we have

$m\left(r,\frac{P[f]}{f^{\gamma_{P}}}\right)=m\left(r,\frac{H_{\nu_{P}}[f]+H_{\nu_{P}+1}[f]+\cdots+H_{\gamma_{P}}[f]}{f^{\gamma_{P}}}\right).$

From the Nevanlinna’s theorem on logarithmic derivative, we get

$m\left(r,\frac{H_{j}[f]}{f^{j}}\right)=S(r,f)~{}\forall~{}j.$

Then

$\displaystyle m\left(r,\frac{P[f]}{f^{\gamma_{P}}}\right)$ $\displaystyle\leq
m\left(r,\frac{H_{\nu_{P}}[f]+H_{\nu_{P}+1}[f]+\cdots+H_{\gamma_{P}-1}[f]}{f^{\gamma_{P}-1}}\right)+m\left(r,\frac{1}{f}\right)+S(r,f)$
$\displaystyle\leq
2~{}m\left(r,\frac{1}{f}\right)+m\left(r,\frac{H_{\nu_{P}}[f]+H_{\nu_{P}+1}[f]+\cdots+H_{\gamma_{P}-2}[f]}{f^{\gamma_{P}-2}}\right)+S(r,f)$
$\displaystyle\vdots$
$\displaystyle\leq(\gamma_{P}-\nu_{P})m\left(r,\frac{1}{f}\right)+S(r,f).$
(21)

Now by the first fundamental theorem of Nevanlinna and (2), we have

$\displaystyle\gamma_{P}~{}T(r,f)$
$\displaystyle=\gamma_{P}~{}m\left(r,\frac{1}{f}\right)+\gamma_{P}~{}N\left(r,\frac{1}{f}\right)+O(1)$
$\displaystyle\leq
m\left(r,\frac{P[f]}{f^{\gamma_{P}}}\right)+m\left(r,\frac{1}{P[f]}\right)+\gamma_{P}~{}N\left(r,\frac{1}{f}\right)+\log{2}+O(1)$
$\displaystyle\leq(\gamma_{P}-\nu_{P})~{}m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{P[f]}\right)+\gamma_{P}~{}N\left(r,\frac{1}{f}\right)+S(r,f)$
$\displaystyle=(\gamma_{P}-\nu_{P})~{}T(r,f)-(\gamma_{P}-\nu_{P})~{}N\left(r,\frac{1}{f}\right)+T(r,P[f])$
$\displaystyle\qquad-N\left(r,\frac{1}{P[f]}\right)+\gamma_{P}~{}N\left(r,\frac{1}{f}\right)+S(r,f).$

That is,

$\nu_{P}~{}T(r,f)\leq\nu_{P}~{}N\left(r,\frac{1}{f}\right)+T(r,P[f])-N\left(r,\frac{1}{P[f]}\right)+S(r,f).$

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

$\displaystyle\nu_{P}~{}T(r,f)$
$\displaystyle\leq\nu_{P}~{}N\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{P[f]}\right)+\overline{N}(r,P[f])$
$\displaystyle\qquad+\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)-N\left(r,\frac{1}{P[f]}\right)+S(r,f)$
$\displaystyle\leq\nu_{P}~{}N\left(r,\frac{1}{f}\right)+\overline{N}(r,f)+\overline{N}\left(r,\frac{1}{P[f]}\right)$
$\displaystyle\qquad+\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)-N\left(r,\frac{1}{P[f]}\right)+S(r,f).$
(22)

We now prove (i).

First note that if $z_{0}$ is a zero of $f$ of multiplicity $m~{}(\geq 1),$
then $P[f]$ has a zero at $z_{0}$ of multiplicity at least
$(m+1)\nu_{P}-\Gamma_{P}$ and hence

$N\left(r,\frac{1}{P[f]}\right)-\overline{N}\left(r,\frac{1}{P[f]}\right)\geq\left[(m+1)\nu_{P}-\Gamma_{P}-1\right]\overline{N}\left(r,\frac{1}{f}\right).$

Then from (2), we obtain

$\displaystyle\nu_{P}~{}T(r,f)$
$\displaystyle\leq\nu_{P}~{}N\left(r,\frac{1}{f}\right)+\overline{N}(r,f)+\left[1+\Gamma_{P}-(m+1)\nu_{P}\right]\overline{N}\left(r,\frac{1}{f}\right)$
$\displaystyle\qquad+\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)+S(r,f).$
$\displaystyle=\overline{N}(r,f)+\left(1+\Gamma_{P}-\nu_{P}\right)\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)+S(r,f).$

Thus

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

This proves (i).

To prove (ii), suppose that $f$ has zeros of multiplicity at least $k+1$ and
$(k+2)\nu_{P}>\Gamma_{P}+2.$ Then one can easily see that

$N\left(r,\frac{1}{P[f]}\right)-\overline{N}\left(r,\frac{1}{P[f]}\right)\geq\left[(k+2)\nu_{P}-\Gamma_{P}-1\right]\overline{N}\left(r,\frac{1}{f}\right).$

Again, from (2), we obtain

$\displaystyle\nu_{P}~{}T(r,f)$
$\displaystyle\leq\nu_{P}~{}N\left(r,\frac{1}{f}\right)+\overline{N}(r,f)+\left[1+\Gamma_{P}-(k+2)\nu_{P}\right]\overline{N}\left(r,\frac{1}{f}\right)$
$\displaystyle\qquad+\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)+S(r,f).$
$\displaystyle\leq\nu_{p}~{}T(r,f)+T(r,f)+\left[1+\Gamma_{P}-(k+2)\nu_{P}\right]T(r,f)$
$\displaystyle\qquad+\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)+S(r,f).$

This implies that

$\left[(k+2)\nu_{P}-\Gamma_{P}-2\right]~{}T(r,f)\leq\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)+S(r,f)$

and hence

$T(r,f)\leq\frac{1}{(k+2)\nu_{P}-\Gamma_{P}-2}\overline{N}\left(r,\frac{1}{P[f]-\psi}\right)+S(r,f).$

##  3 Proofs of Theorems

Proof: [Proof of Theorem 5] Suppose that $\mathcal{F}$ is not normal at
$z_{0}\in D.$ Then by Lemma 1, there exist a sequence
$\left\\{f_{n}\right\\}\subset\mathcal{F},$ a sequence of points
$\left\\{z_{n}\right\\}\subset D$ with $z_{n}\rightarrow z_{0}$ and a sequence
of positive real numbers satisfying $\rho_{n}\rightarrow 0$ such that the
sequence

$\phi_{n}(\zeta):=\frac{f_{n}(z_{n}+\rho_{n}\zeta)}{\rho_{n}^{\alpha}}\rightarrow\phi(\zeta),~{}\alpha:=\Theta_{P}-1,$

locally uniformly in $\mathbb{C},$ where $\phi$ is a non-constant entire
function of finite order with zeros of multiplicity at least $k+1.$ Moreover,
the corresponding sequence $\left\\{g_{n}\right\\}\subset\mathcal{G}$
converges locally uniformly to $g$ in $D,$ where $g~{}(\not\equiv\infty)$ has
zeros of multiplicity at least $k+1.$

Note that

$\displaystyle\tilde{P}[\phi_{n}](\zeta)$
$\displaystyle:=P[f_{n}](z_{n}+\rho_{n}\zeta)$
$\displaystyle=\sum\limits_{i=1}^{m}a_{i}(z_{n}+\rho_{n}\zeta)~{}\rho_{n}^{\left[(1+\alpha)\gamma_{M_{i}}-\Gamma_{M_{i}}\right]}~{}M_{i}[\phi_{n}](\zeta).$

Since

$\alpha=\Theta_{P}-1=\frac{\Gamma_{M_{1}}}{\gamma_{M_{1}}}-1~{}\mbox{ and
}~{}\frac{\Gamma_{M_{1}}}{\gamma_{M_{1}}}>\frac{\Gamma_{M_{t}}}{\gamma_{M_{t}}}~{}\mbox{
for }~{}2\leq t\leq m,$

we have

$\displaystyle\tilde{P}[\phi_{n}](\zeta)$
$\displaystyle=P[f_{n}](z_{n}+\rho_{n}\zeta)$
$\displaystyle=a_{1}(z_{n}+\rho_{n}\zeta)M_{1}[\phi_{n}](\zeta)+\sum\limits_{i=2}^{m}a_{i}(z_{n}+\rho_{n}\zeta)~{}\rho_{n}^{\left[(1+\alpha)\gamma_{M_{i}}-\Gamma_{M_{i}}\right]}~{}M_{i}[\phi_{n}](\zeta).$

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

$\sum\limits_{i=2}^{m}a_{i}(z_{n}+\rho_{n}\zeta)~{}\rho_{n}^{\left[(1+\alpha)\gamma_{M_{i}}-\Gamma_{M_{i}}\right]}~{}M_{i}[\phi_{n}](\zeta)$

converges locally uniformly to $0$ in $\mathbb{C}$ and hence

$\tilde{P}[\phi_{n}](\zeta)=P[f_{n}](z_{n}+\rho_{n}\zeta)\rightarrow
a_{1}(z_{0})M_{1}[\phi](\zeta)$

locally uniformly in $\mathbb{C}.$ Since $a_{1}(z_{0})\neq 0,$ we may assume,
without loss of generality, that $a_{1}(z_{0})=1.$ Also, since
$g_{n}\rightarrow g$ implies that $P[g_{n}]\rightarrow P[g]$ locally uniformly
in $\mathbb{C}.$ Since $\phi$ is an entire function having zeros of
multiplicity at least $k+1,$ $M_{1}[\phi]$ assumes every non-zero value in
$\mathbb{C}.$ In particular, $M_{1}[\phi]$ assumes the non-zero value $a,$
say. Let $\zeta_{0}\in\mathbb{C}$ be such that $M_{1}[\phi](\zeta_{0})=a.$
Since $M_{1}[\phi]\not\equiv a,$ by Hurwitz’s theorem, there exists a sequence
$\zeta_{n}\rightarrow\zeta_{0}$ such that for sufficiently large $n,$

$P[f_{n}](z_{n}+\rho_{n}\zeta_{n})=a$

and so by hypothesis, we have

$P[g_{n}](z_{n}+\rho_{n}\zeta_{n})=a.$

Taking $n\rightarrow\infty,$ we get $P[g](\zeta_{0})=a\neq 0.$

Next, we claim that $\phi$ omits $0.$ For, let $\zeta_{0}\in\mathbb{C}$ be
such that $\phi(\zeta_{0})=0.$ Since $\phi\not\equiv 0,$ by Hurwitz’s theorem,
there exists $\tilde{\zeta_{n}}\rightarrow\zeta_{0}$ such that for
sufficiently large $n,$
$\phi_{n}(\tilde{\zeta_{n}})=\rho_{n}^{-\alpha}f_{n}(z_{n}+\rho_{n}\tilde{\zeta_{n}})=0.$
This implies that $f_{n}(z_{n}+\rho_{n}\tilde{\zeta_{n}})=0$ and so by
hypothesis, $g_{n}(z_{n}+\rho_{n}\tilde{\zeta_{n}})=0.$ Taking
$n\rightarrow\infty,$ we get $g(\zeta_{0})=0.$ Since $g$ has zeros of
multiplicity at least $k+1,$ it follows that $P[g](z_{0})=0,$ a contradiction.
This establishes the claim.

Since $\phi(\zeta)\neq 0,$ $\phi$ is transcendental and so from Lemma 8 (i),
it follows that $M_{1}[\phi]$ assumes $a$ infinitely many times. If
$M_{1}[\phi](\zeta_{0})=a,$ then as done before, we would get
$P[g](\zeta_{0})=a.$ Let $l~{}(\geq 1)$ be the multiplicity of zero of
$P[g]-a$ at $\zeta_{0}.$ Since $P[g]\not\equiv a,$ by Hurwitz’s theorem, for
sufficiently large $n,$ $P[g_{n}]-a$ has exactly $l$ zeros in a neighbourhood
$U(\zeta_{0})$ of $\zeta_{0}.$ Since $M_{1}[\phi]$ assumes $a$ infinitely many
times, we assume that $\zeta_{1},\ldots,\zeta_{l+1}$ be the distinct zeros of
$M_{1}[\phi]-a.$ Again, by Hurwitz’s theorem, for sufficiently large $n,$
there exists $\zeta_{j,n}\rightarrow\zeta_{j}$ such that
$P[\phi_{n}](\zeta_{j,n})=a$ for $j=1,\ldots,l+1.$ Thus there are $l+1$
distinct zeros of $P[\phi_{n}]-a$ in $U(\zeta_{0})$ and hence $P[g_{n}]-a$ has
$l+1$ distinct zeros in $U(\zeta_{0}),$ a contradiction.

Proof: [Proof of Theorem 6] Assume that $\mathcal{F}$ is not normal at
$z_{0}\in D.$ Then by Lemma 1, there exist a sequence
$\left\\{f_{n}\right\\}\subset\mathcal{F},$ a sequence of points
$\left\\{z_{n}\right\\}\subset D$ with $z_{n}\rightarrow z_{0}$ and a sequence
of positive real numbers satisfying $\rho_{n}\rightarrow 0$ such that the
sequence

$\phi_{n}(\zeta):=\frac{f_{n}(z_{n}+\rho_{n}\zeta)}{\rho_{n}^{\alpha}}\rightarrow\phi(\zeta),~{}\alpha:=\Theta_{P}-1,$

locally uniformly in $\mathbb{C},$ where $\phi$ is a non-constant meromorphic
function of finite order with zeros of multiplicity at least $k+1.$ Moreover,
the corresponding sequence $\left\\{g_{n}\right\\}\subset\mathcal{G}$
converges locally uniformly to $g$ in $D,$ where $g~{}(\not\equiv\infty)$ has
zeros of multiplicity at least $k+1.$ Then by a similar argument as in the
proof of Theorem 5, we find that on every compact subset of $\mathbb{C}$ not
containing poles of $\phi,$

$\tilde{P}[\phi_{n}](\zeta):=P[f_{n}](z_{n}+\rho_{n}\zeta)\rightarrow
a_{1}(z_{0})M_{1}[\phi](\zeta)~{}\mbox{as }n\rightarrow\infty.$

Also, for every $\zeta\in\mathbb{C}\setminus\left\\{g^{-1}(\infty)\right\\},$
we have $P[g_{n}](\zeta)\rightarrow P[g](\zeta)~{}\mbox{as
}n\rightarrow\infty$ locally uniformly. Since
$M_{1}[\phi]=\prod\limits_{j=0}^{k}\left(\phi^{(j)}\right)^{n_{j1}},$ we shall
denote, for the sake of convenience, $M_{1}[\phi]$ by $M[\phi]$ and the powers
$n_{j1}$ by $n_{j}$ for $j=1,\ldots,k.$ Also, we may assume that
$a_{1}(z_{0})=1.$ Now, we consider the following cases:

Case 1: $n_{0}=n_{1}=\cdots=n_{k-1}=0.$ In this case, we have
$M[\phi]=\left(\phi^{(k)}\right)^{n_{k}},$ where $n_{k}\geq 1.$
Claim 1: $\phi(\zeta)\neq 0.$

For, let $\zeta_{0}\in\mathbb{C}$ be such that $\phi(\zeta_{0})=0.$ Since
$\phi\not\equiv 0,$ by Hurwitz’s theorem, there exists
$\zeta_{n}\rightarrow\zeta_{0}$ such that for sufficiently large $n,$
$\phi_{n}(\zeta_{n})=\rho_{n}^{-\alpha}f_{n}(z_{n}+\rho_{n}\zeta_{n})=0.$ This
implies that $f_{n}(z_{n}+\rho_{n}\zeta_{n})=0$ and so by hypothesis,
$g_{n}(z_{n}+\rho_{n}\zeta_{n})=0.$ Taking $n\rightarrow\infty,$ we get
$g(z_{0})=0.$ Since $g$ has zeros of multiplicity at least $k+1,$ it follows
that $P[g](z_{0})=0.$ If $\phi$ is transcendental, then by Lemma 3, $M[\phi]$
assumes the non-zero complex number $a.$ Let $\zeta^{\prime}\in\mathbb{C}$
such that $M[\phi](\zeta^{\prime})=a.$ Since $M[\phi]\not\equiv a,$ by
Hurwitz’s theorem, there exists a sequence
$\zeta_{n}^{\prime}\rightarrow\zeta^{\prime}$ such that for sufficiently large
$n,$ $P[f_{n}](z_{n}+\rho_{n}\zeta_{n}^{\prime})=a$ and hence by hypothesis,
we get $P[g_{n}](z_{n}+\rho_{n}\zeta_{n}^{\prime})=a.$ Taking
$n\rightarrow\infty,$ we get $P[g](z_{0})=a\neq 0,$ a contradiction. Hence
$\phi$ must be rational. Furthermore, using the preceding argument, one can
easily see that $\phi$ is not a polynomial and therefore there exists
$\zeta^{*}\in\mathbb{C}$ such that $\phi(\zeta^{*})=\infty.$ Again, by
Hurwitz’s theorem, there exists a sequence $\zeta_{n}^{*}\rightarrow\zeta^{*}$
such that for sufficiently large $n,$ we have
$f_{n}(z_{n}+\rho_{n}\zeta_{n}^{*})=\infty$ and hence
$g_{n}(z_{n}+\rho_{n}\zeta_{n}^{*})=\infty.$ Taking $n\rightarrow\infty,$ we
get $g(z_{0})=\infty,$ which is again a contradiction. Hence $\phi(\zeta)\neq
0.$
Claim 2: $\phi$ has no poles.

Suppose there is some $\zeta_{1}\in\mathbb{C}$ such that
$\phi(\zeta_{1})=\infty.$ Then as done before, we get $g(z_{0})=\infty.$ Since
$\phi(\zeta)\neq 0,$ by Lemma 2, there is some $\zeta_{2}\in\mathbb{C}$ such
that $M[\phi](\zeta_{2})=a.$ Again, by Hurwitz’s theorem, there exists a
sequence $\zeta_{n,2}\rightarrow\zeta_{2}$ such that for sufficiently large
$n,$ we have $P[f_{n}](z_{n}+\rho_{n}\zeta_{n,2})=a.$ This implies that
$P[g_{n}](z_{n}+\rho_{n}\zeta_{n,2})=a$ and hence $P[g](z_{0})=a$ which is not
possible since $P[g]\not\equiv a$ and $g(z_{0})=\infty.$ This establishes
Claim 2.

Now, Claim 1 and Claim 2 together imply that $\phi$ is a non-vanishing
transcendental entire function and so from Lemma 2, it follows that $M[\phi]$
assumes $a$ infinitely many times. Now, proceeding similarly as in the proof
of Theorem 5, we get a contradiction.

Case 2: $n_{0}\geq 1,~{}n_{k}\geq 1.$ Then $\gamma_{M}\geq 2$ and a simple
computation shows that

$\displaystyle(k+2)\gamma_{M}-\Gamma_{M}-2$
$\displaystyle=(k+2)(n_{0}+n_{1}+\cdots+n_{k})-[n_{0}+2n_{1}+\cdots+(k+1)n_{k}]-2$
$\displaystyle=(k+2-1)n_{0}+(k+2-2)n_{1}+\cdots+[k+2-(k+1)]n_{k}-2$
$\displaystyle\geq kn_{0}+(k+2-2)n_{1}+\cdots+[(k+2)-(k-1)]n_{k-1}$
$\displaystyle\geq 1.$

Now using the same arguments as that of Case 1 and Lemma 8 (ii), we obtain
$\phi(\zeta)\neq 0$ and hence $\phi$ is not a polynomial. If $\phi$ is
transcendental or rational, then we use Lemma 8 (i), and the arguments from
Claim 2 to conclude that $\phi$ is a non-vanishing transcendental entire
function. Thus from Lemma 8 (i), it follows that $M[\phi]$ assumes $a$
infinitely many times. Again, proceeding similarly as in the proof of Theorem
5, we get a contradiction. Hence $\mathcal{F}$ is normal in $D.$

Proof: [Proof of Theorem 8] Suppose that $\mathcal{F}$ is not normal at
$z_{0}\in D.$ Then we consider the following two cases:

Case 1: $h(z_{0})\neq 0.$ By Lemma 1, there exist a sequence
$\left\\{f_{j}\right\\}\subset\mathcal{F},$ a sequence of points
$\left\\{z_{j}\right\\}\subset D$ with $z_{j}\rightarrow z_{0}$ and a sequence
of positive real numbers satisfying $\rho_{j}\rightarrow 0$ such that the
sequence

$g_{j}(\xi):=\frac{f_{j}(z_{j}+\rho_{j}\xi)}{\rho_{j}^{\alpha}}\rightarrow
g(\xi),~{}\alpha:=\frac{\Gamma_{M}}{\gamma_{M}}-1,$

locally uniformly in $\mathbb{C}$ with respect to the spherical metric, where
$g\in\mathcal{M}(\mathbb{C})$ is a non-constant function of order at most $2.$
Since each $f_{j}$ has zeros of multiplicity at least $k+m+1,$ by Argument
principle, it follows that $g$ has zeros of multiplicity at least $k+m+1.$
Furthermore, a simple computation shows that

$\displaystyle M[g_{j}](\xi)$
$\displaystyle=\prod\limits_{i=0}^{k}\left(\left(\frac{f_{j}(z_{j}+\rho_{j}\xi}{\rho_{j}^{\alpha}}\right)^{(i)}\right)^{n_{i}}$
$\displaystyle=\frac{\rho_{j}^{n_{1}+2n_{2}+\cdots+kn_{k}}}{\rho_{j}^{\alpha(n_{0}+n_{1}+\cdots+n_{k})}}\prod\limits_{i=0}^{k}\left(\left(f_{j}(z_{j}+\rho_{j}\xi)\right)^{(i)}\right)^{n_{i}}$
$\displaystyle=\frac{\rho_{j}^{\Gamma_{M}-\gamma_{M}}}{\rho_{j}^{\alpha\gamma_{M}}}M[f_{j}](z_{j}+\rho_{j}\xi)$
$\displaystyle=M[f_{j}](z_{j}+\rho_{j}\xi).$

Hence on every compact subset of $\mathbb{C}$ not containing poles of $g,$ we
have

$M[g_{j}](\xi)=M[f_{j}](z_{j}+\rho_{j}\xi)\rightarrow M[g](\xi).$

Obviously, $M[g](\xi)\not\equiv h(z_{0}).$ Also, one can easily check that
$(k+2)\gamma_{M}-\Gamma_{M}-2>0.$ Thus by Lemma 5 and Lemma 8 (ii),
$M[g](\xi)-h(z_{0})$ has at least two distinct zeros, say $\xi_{1}$ and
$\xi_{2}.$ Choose $\delta>0$ small enough such that $D(\xi_{1},\delta)\cap
D(\xi_{2},\delta)=\phi$ and for any $\xi\neq\xi_{1},~{}\xi_{2}$ in
$D(\xi_{1},\delta)\cup D(\xi_{2},\delta),$ $M[g](\xi)-h(z_{0})\neq 0.$ By
Hurwitz’s theorem, there exist sequences $\xi_{j,1}\rightarrow\xi_{1}$ and
$\xi_{j,2}\rightarrow\xi_{2}$ such that for sufficiently large $j,$

$M[f_{j}](z_{j}+\rho_{j}\xi_{j,l})-h(z_{j}+\rho_{j}\xi_{l})=0,~{}l=1,~{}2.$

By hypothesis, it follows that for any fixed integer $p$ and for all $j,$

$M[f_{p}](z_{j}+\rho_{j}\xi_{j,l})-h(z_{j}+\rho_{j}\xi_{l})=0,~{}l=1,~{}2.$

Taking $j\rightarrow\infty,$ we get

$M[f_{p}](z_{0})-h(z_{0})=0.$

Since the zeros of $M[f_{p}]-h(z_{0})$ have no accumulation point, for
sufficiently large $j,$ we obtain
$z_{j}+\rho_{j}\xi_{j,1}=0=z_{j}+\rho_{j}\xi_{j,2}.$ This implies that
$\xi_{j,1}=-z_{j}/\rho_{j}=\xi_{j,2}$ showing that $D(\xi_{1},\delta)\cap
D(\xi_{2},\delta)\neq\phi,$ a contradiction.

Case 2: $h(z_{0})=0.$ Then $h(z)=z^{t}h_{1}(z),$ where $h_{1}$ is holomorphic
in $D$ such that $h_{1}(z_{0})\neq 0$ and $1\leq t\leq m.$ We may assume that
$h_{1}(z_{0})=1.$ Again, by Lemma 1, there exist a sequence
$\left\\{f_{j}\right\\}\subset\mathcal{F},$ a sequence of points
$\left\\{z_{j}\right\\}\subset D$ with $z_{j}\rightarrow z_{0}$ and a sequence
of positive real numbers satisfying $\rho_{j}\rightarrow 0$ such that the
sequence

$g_{j}(\xi):=\frac{f_{j}(z_{j}+\rho_{j}\xi)}{\rho_{j}^{\beta}}\rightarrow
g(\xi),~{}\beta:=\frac{\Gamma_{M}+t}{\gamma_{M}}-1,$

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

Subcase 2.1: Suppose that there exists a subsequence of $z_{j}/\rho_{j}$
(which, for the sake of convenience, is again denoted by $z_{j}/\rho_{j}$)
such that $z_{j}/\rho_{j}\rightarrow\infty$ as $j\rightarrow\infty.$

Consider the family

$V_{j}(\xi):=\frac{f_{j}(z_{j}+z_{j}\xi)}{z_{j}^{\beta}}.$

By simple calculation, one can see that
$M[f_{j}](z_{j}+z_{j}\xi)=z_{j}^{t}M[V_{j}](\xi).$ By hypothesis, for any
distinct integers $r,~{}s,$ we have,

$M[f_{r}](z)-h(z)=0\Leftrightarrow~{}M[f_{s}](z)-h(z)=0.$

Choose $\xi,$ say $\xi_{i}$ such that for some $j,$ say $j_{i},$
$z_{j_{i}}+z_{j_{i}}\xi_{i}$ is a zero of $M[f_{r}](z)-h(z)$ and hence of
$M[f_{s}](z)-h(z).$ Then

$\displaystyle
M[f_{r}](z_{j_{i}}+z_{j_{i}}\xi_{i})=h(z_{j_{i}}+z_{j_{i}}\xi_{i})\Leftrightarrow~{}M[f_{s}](z_{j_{i}}+z_{j_{i}}\xi_{i})=h(z_{j_{i}}+z_{j_{i}}\xi_{i})$
$\displaystyle\Rightarrow
M[V_{r}](\xi_{i})=(1+\xi_{i})^{t}h_{1}(z_{j_{i}}+z_{j_{i}}\xi_{i})\Leftrightarrow~{}M[V_{s}](\xi_{i})=(1+\xi_{i})^{t}h_{1}(z_{j_{i}}+z_{j_{i}}\xi_{i}).$

Since
$(1+\xi_{i})^{t}h_{1}(z_{j_{i}}+z_{j_{i}}\xi_{i})\rightarrow(1+\xi_{i})^{t}\neq
0$ in a sufficiently small neighbourhood $N_{0}$ of the origin, it follows
that $(1+\xi_{i})^{t}h_{1}(z_{j_{i}}+z_{j_{i}}\xi_{i})\neq 0$ in $N_{0}.$ Thus
by Case 1, $\left\\{V_{j}\right\\}$ is normal in $N_{0}$ and so there exists a
subsequence of $\left\\{V_{j}\right\\}$ (again denoted by
$\left\\{V_{j}\right\\}$) such that $V_{j}$ converges spherically locally
uniformly in $N_{0}$ to a meromorphic function $V.$

Subcase 2.1.1: $V(0)\neq 0.$ Then for any $\xi\in\mathbb{C},$

$g_{j}(\xi)=\rho_{j}^{-\beta}f_{j}(z_{j}+\rho_{j}\xi)=\left(\frac{z_{j}}{\rho_{j}}\right)^{\beta}V_{j}\left(\frac{\rho_{j}}{z_{j}}\xi\right)$

converges spherically locally uniformly to $\infty,$ showing that
$g\equiv\infty,$ a contradiction.

Subcase 2.1.2: $V(0)=0.$ Then $V^{\prime}(0)\neq\infty$ and so

$g_{j}^{\prime}(\xi)=\rho_{j}^{-\beta+1}f_{j}^{\prime}(z_{j}+\rho_{j}\xi)=\left(\frac{\rho_{j}}{z_{j}}\right)^{-\beta+1}V_{j}^{\prime}\left(\frac{\rho_{j}}{z_{j}}\xi\right)$

converges spherically locally uniformly in $\mathbb{C}$ to $0.$ This shows
that $g^{\prime}\equiv 0,$ a contradiction to the fact that $g$ is non-
constant.

Subcase 2.2: Suppose there exists a subsequence of $z_{j}/\rho_{j}$ (which,
for the sake of convenience, is again denoted by $z_{j}/\rho_{j}$) such that
$z_{j}/\rho_{j}\rightarrow\zeta$ as $j\rightarrow\infty,$ where
$\zeta\in\mathbb{C}.$

Consider the family $W_{j}(\xi):=\rho_{j}^{-\beta}f_{j}(\rho_{j}\xi).$ Then

$W_{j}(\xi)=g_{j}\left(\xi-\frac{z_{j}}{\rho_{j}}\right)\rightarrow
g(\xi-\zeta):=W(\xi).$

It easily follows that on any compact subset of $\mathbb{C},$ containing no
poles of $W,$

$M[W_{j}](\xi)-\xi^{t}\rightarrow M[W](\xi)-\xi^{t}.$

Clearly, $M[W](\xi)\not\equiv\xi^{t}.$ Thus by Lemma 7, $M[W](\xi)-\xi^{t}$
has at least two distinct zeros. Proceeding similarly as in Case 1, we get a
contradiction. Hence $\mathcal{F}$ is normal in $D.$

##  4 Statements and Declarations

  * 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.

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

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# The Hidden Phase of the Spin-Boson Model

Florian Otterpohl  Center for Computational Quantum Physics, Flatiron
Institute, New York, New York 10010, USA  I. Institut für Theoretische Physik,
Universität Hamburg, Notkestr. 9, 22607 Hamburg, Germany
Peter Nalbach  Fachbereich Wirtschaft & Informationstechnik, Westfälische
Hochschule, Münsterstr. 265 46397 Bocholt, Germany  Michael Thorwart  I.
Institut für Theoretische Physik, Universität Hamburg, Notkestr. 9, 22607
Hamburg, Germany

###### Abstract

A quantum two-level system immersed in a sub-Ohmic bath experiences enhanced
low-frequency quantum statistical fluctuations which render the nonequilibrium
quantum dynamics highly non-Markovian. Upon using the numerically exact time-
evolving matrix product operator approach, we investigate the phase diagram of
the polarization dynamics. In addition to the known phases of damped coherent
oscillatory dynamics and overdamped decay, we identify a new third region in
the phase diagram for strong coupling showing an aperiodic behaviour. We
determine the corresponding phase boundaries. The dynamics of the quantum two-
state system herein is not coherent by itself but slaved to the oscillatory
bath dynamics.

††preprint: APS

Introduction. —Dissipative environments are the cause of relaxation and
decoherence in quantum systems. Accordingly, understanding and modelling their
influence and subsequently tailoring their impact is relevant to many research
areas [1, 2]. At strong system-bath coupling, dissipative environments may
also lead to fully incoherent dynamics or even complete suppression of the
dynamics (localization). Generally it is believed that, while prominent
environmental modes may well cause coherent oscillations in a quantum system,
a broadband reservoir destroys coherence at sufficiently strong coupling [3,
4].

Typically, a two-state system interacting with harmonic degrees of freedom
(spin-boson model) is studied [1, 2]. At low temperatures and weak coupling
damped coherent oscillations are observed while at stronger dissipation a
classical incoherent decay towards thermal equilibrium is found. For an Ohmic
bath with spectrum $J(\omega)\propto\alpha\omega^{s}$ and spectral exponent
$s=1$ the ratio of damping rate and oscillation frequency is independent of
the oscillation frequency. With increasing coupling $\alpha$ the dynamics
turns incoherent and, for even larger coupling, a zero temperature phase
transition towards a localized phase is observed [1, 2].

While super-Ohmic spectra with $s>1$ with more pronounced high-frequency modes
do not show incoherent dynamics nor a localization transition, reservoirs with
more pronounced low frequency spectra such as sub-Ohmic ones with $s<1$
exhibit a localization phase transition [5, 6, 7] and are relevant for quantum
impurity systems [8, 9]. In contrast, the dynamic transition to incoherent
behaviour is far less studied although this class of reservoirs constitutes
the dominant noise source in, for example, nanomechanical oscillators systems
[10] and superconducting qubit architectures (specifically charge noise
generated by two-level fluctuators) [11, 12, 13]. Although originally believed
to be always incoherent [1], it is now well established that for $1/2\lesssim
s\leq 1$ with increasing coupling the dynamics turns from damped oscillatory
to incoherent, and for larger coupling a transition to a localized phase takes
place [14, 15, 16, 17]. For $0\leq s\lesssim 1/2$, however, Kast and Ankerhold
[16] showed that oscillatory dynamics persists for arbitrary coupling strength
for an initially polarized bath. Pure dephasing sub-Ohmic reservoirs influence
a quantum two-state system in a similar way and result in overdamping only for
$\omega_{c}\rightarrow\infty$ and $s\gtrsim 0.1$ [18].

This leaves two important questions: What happens when crossing the supposed
phase boundary at $s\simeq 1/2$ for a fixed but strong coupling and why can
strong sub-Ohmic fluctuations not turn the system dynamics incoherent? We
investigate for $1/2\lesssim s\leq 1$ and increasing coupling to the
environment the dynamics which turns from coherent to incoherent for a
coupling $\alpha_{D}(s)$ which depends on the spectral exponent $s$. For even
larger couplings we encounter a sharp phase boundary at a coupling
$\alpha_{B}(s)$ to aperiodic behaviour, i.e., the dynamics is not fully
incoherent but exhibits a single turnaround. This new phase extends to all
$0\leq s\leq 1$. For $0\leq s\lesssim 1/2$, where Ref. [16] reports damped
oscillatory dynamics for all couplings, we find with increasing coupling a
sharp transition from oscillatory dynamics with various minima and maxima to
aperiodic dynamics with a single minimum for an initially unpolarized bath.
The frequency related to the minimum is proportional to the reservoir cut-off
frequency. Accordingly, this dynamics is not generated by the central quantum
system but by the reservoir and it turns incoherent for
$\omega_{c}\rightarrow\infty$. We simulate the dynamics in a broad range of
parameters in order to map the new phase diagram of the dynamical behaviour of
the spin-boson model.

Model. —We consider the symmetric spin-boson model ($\hbar=1$, $k_{B}=1$) with
the Hamiltonian

$\displaystyle\hat{H}$
$\displaystyle=\hat{H}_{S}+\hat{H}_{B}+\hat{H}_{\text{int}}$ (1)
$\displaystyle=\frac{\Omega}{2}\hat{\sigma}_{x}+\frac{1}{2}\sum_{j}\left(\hat{p}_{j}^{2}+\omega_{j}\hat{x}_{j}^{2}\right)+\frac{\hat{\sigma}_{z}}{2}\hat{\xi},$

where $\hat{\xi}=\sum_{j}c_{j}\hat{x}_{j}$, $\Omega$ is the tunneling
splitting, and $\hat{\sigma}_{x/z}$ are the Pauli matrices. The bath consists
of harmonic oscillators with momenta $\hat{p}_{j}$, angular frequencies
$\omega_{j}$, and positions $\hat{x}_{j}$ which are coupled via coupling
constants $c_{j}$ to the spin. The bath has the spectral density

$J(\omega)=\sum_{j}\frac{c_{j}^{2}}{2\omega_{j}}\delta(\omega-\omega_{j})=2\alpha\frac{\omega^{s}}{\omega_{c}^{s-1}}e^{-\omega/\omega_{c}},$
(2)

with the high-frequency cut-off $\omega_{c}=10\Omega$ (unless specified
otherwise) and the coupling strength $\alpha$. We focus on the (sub-)Ohmic
regime where $0\leq s\leq 1$.
We calculate the time-dependent polarization $P(t)=\Braket{\sigma_{z}}(t)$,
using a factorizing initial preparation of the system with $P(0)=1$ and the
thermal distribution of the initially uncoupled (i.e., unpolarized) bath at
zero temperature.

Method. — To determine $P(t)$ we use the numerically exact real-time
quasiadiabatic propagator path integral (QUAPI) [19, 20, 21, 22, 23, 24]. Once
the bath oscillators have been integrated out, an effective dynamics of the
system arises which is nonlocal in time. To treat the highly entangled system-
bath dynamics of the sub-Ohmic spin-boson model, we make use of the time-
evolving matrix product operator (TEMPO) technique [23, 24], see Supplemental
Material for further details. It allows us to avoid the conventional QUAPI
memory cutoff as it rewrites the arising discrete path-sum in terms of a
numerically highly efficient tensor network.

Polarization dynamics. — At first, we investigate the dynamics for a fixed
$s=0.7$ for increasing coupling $\alpha$. Fig. 1 shows the time-dependent
polarization $P(t)$ for times up to $\Omega t\lesssim 10$. For $\alpha=0.05$
and $\alpha=0.15$, we observe damped oscillatory or coherent dynamics with a
minimum (marked by a red cross in Fig. 1) and a maximum (marked by a green
diamond) visible in the studied time frame. But for couplings $\alpha=0.2$ and
$\alpha=0.4$, we find a monotonic decay highlighting purely incoherent
dynamics. Surprisingly, for larger couplings $\alpha\geq 0.5$ we find again a
minimum but no subsequent maximum. We denote this dynamics pseudo-coherent in
the following.

Figure 1: Polarization $P(t)$ for $T=0$, $s=0.7$, and different couplings
strengths $\alpha$. The arrow intersects the lines in ascending order of
coupling strengths. Local minima (maxima) are marked by a red cross (green
diamond).

Next we decrease the spectral exponent $s$ for a strong but fixed system bath
coupling strength $\alpha=0.8$ with dynamics in the pseudo-coherent phase from
$s=0.7$ to $0.3$. Fig. 2 depicts the according time-dependent polarization
$P(t)$. In all cases we observe pseudo-coherent dynamics with a single minimum
(marked by a red cross). With decreasing $s$, the minimum shifts to earlier
times. No qualitative change in the dynamic behaviour is found upon decreasing
$s$ below $s=1/2$. For $s\lesssim 1/2$, Kast and Ankerhold [16] have shown
that no transition to incoherent dynamics occurs irrespective of the coupling
strength.

Figure 2: Polarization $P(t)$ for $T=0$, $\alpha=0.8$, and different spectral
exponents $s$. The arrow intersects the lines in ascending order of spectral
exponents. Local minima are marked with a red cross.

Fig. 3 shows the polarization for various coupling strengths for a fixed
$s=0.3$. For couplings from $\alpha=0.8$ down to $\alpha=0.125$, we observe
pseudo-coherent dynamics with only one minimum. For $\alpha=0.1$ and
$\alpha=0.05$, we find coherent dynamics with a minimum and a maximum.

Phase diagram. — Next we examine the full parameter space of $0\leq s\leq 1$
and $\alpha$. We observe sharp transitions between three dynamical phases:
coherent dynamics, i.e. damped oscillatory behaviour with minima and maxima,
incoherent dynamics, i.e. purely monotonic decay, and pseudo-coherent
dynamics, i.e. a single minimum and subsequent decay into localization. The
observed phase diagram is sketched in Fig. 4. For $\alpha=0$, the time-
dependent polarization is $P(t)=\cos(\Omega t)$ with its first local minimum
at $\Omega t=\pi$. As the dynamics depends continuously on the parameters
$\alpha$ and $s$, we can track this local minimum while increasing the
coupling strength. For $s\gtrsim 0.45$, we find that this minimum vanishes if
the coupling strength is sufficiently increased leading to the transition from
coherent to incoherent dynamics at $\alpha_{D}(s)$ (depicted as blue crosses
in Fig. 4). Our results coincide with results shown in Fig. 5 of Ref. [17] for
$s>0.5$. Increasing the coupling strength further, we enter the pseudo-
coherent phase at $\alpha_{B}(s)$ (depicted by orange diamonds). For $s\leq
0.45$ we track the first local maximum of the coherent phase, which vanishes
in the transition to the pseudo-coherent regime (see Fig. 3 for $s=0.3$). The
green circles in Fig. 4 depict the coupling strength $\alpha_{B}(s)$ of the
respective phase transition from coherent to pseudo-coherent dynamics. Note
that while maxima in the dynamics of the pseudo-coherent phase at very long
times, i.e. time-scales not determined by $\Omega$ nor $\omega_{c}$ (which we
cannot strictly rule out due to finite simulation times), would add to the
characteristics of the pseudo-coherent phase, they would not change the phase
diagram which is determined by sharp transitions between different dynamic
behaviour at times well within our simulation time window.

Figure 3: Polarization $P(t)$ for $T=0$, $s=0.3$, and different coupling
strengths $\alpha$. The arrow intersects the lines in ascending order of
coupling strengths. Local minima (maxima) are marked by a red cross (green
diamond).

Pseudo-coherent phase. — Next, we investigate the pseudo-coherent phase in
more detail. Fig. 5 shows the quantity $1-P(t)$ versus the rescaled time
$\omega_{c}t$ for various $\omega_{c}$ for $s=0.3$ and $\alpha=0.8$. (Note
that the minima in $P(t)$ are maxima in this plot). We observe that the
maximum in $1-P(t)$ (corresponding to a minimum in $P(t)$) for sufficiently
large values of $\omega_{c}\gtrsim 10\Omega$ occurs at a time $t_{\rm
min}\simeq\text{const.}/\omega_{c}$. Hence, the observed oscillatory behaviour
in this phase is purely bath-driven and therefore distinct from coherence,
which generally refers to a behaviour inherent to the system. This furthermore
suggests that for $\omega_{c}\rightarrow\infty$ the polarization $P(t)$
approaches 1 and the minimum is attained at $t_{\rm min}\rightarrow 0$. Thus,
only incoherent fully localized behaviour emerges, i.e., the observed
oscillatory behaviour in the pseudo-coherent phase is a result of the finite
bath reaction time of $\mathcal{O}(1/\omega_{c})$ only after which the system
is drawn into localization. Similarly, purely dephasing sub-Ohmic reservoirs
for small $s$ renormalize for increasing $\omega_{c}$ the oscillatory
frequency always to larger values than its damping rate and the system
frequency component in the total renormalized frequency vanishes in that
regime [18]. Thus, the dynamics never becomes fully overdamped.

Because the oscillation frequency is renormalized in the case of a polarized
bath that is initially equilibrated to a spin pointing up, the oscillatory
behavior in Ref. [16] also shows maxima in the pseudo-coherent phase.
Nevertheless, the dynamics in that case is generated by the bath in the same
way (for details, see Supplemental Material).

Figure 4: Phase diagram of the (sub-)Ohmic spin-boson model at $T=0$ and
$\omega_{c}=10\Omega$. The symbols represent coupling strengths $\alpha(s)$ at
which a transition occurs and are linearly interpolated for better visibility.
Blue Cross: The first local minimum vanishes when increasing coupling out of
the coherent to the incoherent domain. Orange diamond: A local minimum appears
for $\Omega t<8$ when increasing coupling strength out of the incoherent
domain. Green circle: The first local maximum vanishes when going from the
coherent to the pseudo-coherent domain.

Increasing $\omega_{c}$ also changes the phase separation line
$\alpha_{B}(s)$. To determine the phase separation line $\alpha_{B}(s)$, we
simulate the polarization dynamics to roughly similar times, i.e., $\Omega
t\simeq 10$ for every $\omega_{c}$. Thus, we are here limited to a fairly
small regime of $\omega_{c}\leq 50\Omega$ to ensure numerical accuracy. For
all studied values of $\omega_{c}$ we find that $\alpha_{B}(s)>\alpha_{c}(s)$
with the critical coupling strength $\alpha_{c}(s)$ for the thermodynamic
$T=0$ phase transition to localization. We find that $\alpha_{B}(s)$ decreases
with increasing $\omega_{c}$.

Our investigation allows to conclude that for $\omega_{c}\to\infty$ the
pseudo-coherent phase turns fully incoherent. Thus, for $s\gtrsim 0.45$ it
seems reasonable that $\alpha_{B}(s,\omega_{c})\to\alpha_{c}(s,\omega_{c})$
since for couplings smaller than $\alpha_{B}(s,\omega_{c})$ the dynamics is
already incoherent for all $\omega_{c}$. For $s\lesssim 0.45$ the situation is
less clear and it remains open whether for $\omega_{c}\to\infty$ there is a
coupling regime with damped coherent dynamics. We should point out that the
limit of infinite $\omega_{c}$ is also not clear for $\alpha_{c}$, i.e. the
thermodynamic localization transition, except for Ohmic bath spectra where
$\alpha_{c}(s=1,\omega_{c}\to\infty)=1$. For $0\leq s\leq 1/2$ a variational
approach yields $\alpha_{c}(s,\omega_{c}\to\infty)\to 0$ [25].

Figure 5: Polarization $1-P(t)$ for $T=0$, $\alpha=0.8$, $s=0.3$, and
different bath cut-off frequencies $\omega_{c}$. The arrow intersects the
lines in ascending order of cut-off frequencies. Local maxima are marked with
a green diamond. Note that the $\omega_{c}=10\Omega$ line corresponds to the
$s=0.3$ line in Fig. 2.

Spectral crossover. — Investigations of the thermodynamic localization phase
transition [6], i.e. the according critical exponents, conclude that $s=1/2$
is the border between mean-field behaviour for $s<1/2$ and non mean-field for
$s>1/2$. We find a crossover $s_{\rm cross}$ with a possible incoherent phase
for $s>s_{\rm cross}$ and no incoherent phase for $s<s_{\rm cross}$ at $s_{\rm
cross}\simeq 0.45$ (see Fig. 4), which is consistent with the results of Duan
et al. [17] that imply $s_{\rm cross}\lesssim 0.525$. Fig. 6 shows the
polarization for $T=0$, $s=0.45$, and different coupling strengths $\alpha$.
For $\alpha=0.12$, we observe coherent dynamics with a minimum and maximum.
For increasing couplings, i.e. $\alpha=0.14$ and $0.157$, both merge. For
$\alpha=0.16$, a new single minimum at latest observed time appears which for
further increasing coupling shifts to earlier times. For $s<0.45$ (see, for
example, Fig. 3), the observed minimum in the pseudo-coherent phase emerges
from the first minimum in the coherent phase. It remains an open question how
this crossover $s$ depends on $\omega_{c}$ and in how far it is connected to
the crossover from mean-field to non-mean-field type thermodynamics phase
behaviour.

Figure 6: Polarization $P(t)$ for $T=0$, $s=0.45$, and different coupling
strengths $\alpha$. The arrow intersects the lines in ascending order of
coupling strengths. Local minima (maxima) are marked with a red cross (green
diamond).

Conclusion. — By means of numerically exact real-time path integral
simulations, we have studied the nonequilibrium dynamics of the (sub-)Ohmic
spin-boson model. Analysing the polarization dynamics, we find for all
spectral exponents $0\leq s\leq 1$ at strong coupling a pseudo-coherent phase
whose hallmark is a single oscillatory minimum. In particular for $0.45\leq
s\leq 1$, we observe this new dynamical phase at strong coupling beside the
well-known coherent dynamics at weak coupling and incoherent dynamics at
intermediate coupling. Oscillatory dynamics for $0\leq s\leq 1/2$ for all
couplings has been observed by Kast and Ankerhold [16] before. We show that
for these $s$ there is nevertheless a transition from coherent (many minima
and maxima in the polarization dynamics) to pseudo-coherent dynamics (with a
single minimum). The frequency related to the oscillatory minimum in the
pseudo-coherent phase is proportional to the bath cut-off frequency.
Accordingly, this dynamics is not generated by the two-level system but by the
reservoir and it turns incoherent for $\omega_{c}\rightarrow\infty$. We map
the full dynamical phase diagram with now three distinct phases.

F.O. is grateful for the kind hospitality at the Flatiron Institute where
parts of this work have been carried out.

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