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\end{document}


2021

[2]Rahul Kumar

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

[2]Department of Mathematics, University of Jammu, Jammu, 180006, Jammu and
Kashmir, India

# Counterexamples to Bloch’s Principle and its Converse in Several Complex
Variables

Kuldeep Singh Charak<EMAIL_ADDRESS><EMAIL_ADDRESS>[  *

###### Abstract

In this paper, besides a counterexample to Bloch’s principle, normality
criteria leading to counterexamples to the converse of Bloch’s principle in
several complex variables are proved. Some Picard-type theorems and their
corresponding normality criteria in $\mathbb{C}^{n}$ are also obtained.

###### keywords:

Normal families, Picard type theorems, Bloch’s principle

###### pacs:

[

MSC Classification]32A19; 32A10

††footnotetext: The work of the second author is supported by CSIR, India
(File No. 09/100(0217)/2018-EMR-I).

This paper is divided into the following four sections:

  1. 1.

Introduction and Auxiliary Results;

  2. 2.

Counterexamples to Bloch’s Principle and its Converse;

  3. 3.

Picard-type Theorems and their Corresponding Normality Criteria;

  4. 4.

Extension of some known Results from $\mathbb{C}$ to $\mathbb{C}^{n}$.

##  1 Introduction and Auxiliary Results

Let $D$ to be a domain in $\mathbb{C}^{n}$ and $\mathcal{F}$ to be a family of
holomorphic functions in $D$. $\mathcal{F}$ is said to be normal in $D$ if
each sequence in $\mathcal{F}$ contains a subsequence that converges locally
uniformly in $D.$ The aim of this paper is to obtain normality criteria
leading to counterexamples to the converse of Bloch’s principle in several
complex variables. Bloch’s principle in one complex variable is extensively
studied and the reader may refer to Robinson ; Rubel ; Zalcman ; Bergweiler ;
Li ; Lahiri ; Charak1 ; Charak2 ; Charak3 , in order. Bloch’s principle states
that a family $\mathcal{F}$ of holomorphic functions of several complex
variables in a domain $D\subset\mathbb{C}^{n}$ satisfying a certain property
$\bf P$ in $D$ is likely to be normal in $D$ if any entire function possessing
the property $\bf P$ in $\mathbb{C}^{n}$ reduces to a constant. In one complex
variable neither Bloch’s principle nor its converse holds in general, for
example one may refer to Rubel ; Lahiri . For the construction of
counterexamples to the converse, one needs to find a normality criterion with
a certain property and then find out a non-constant entire function satisfying
this property in $\mathbb{C}^{n}$; this sounds interesting particularly from
the experience of one complex variable case and this is what we have explored
in the present paper in several complex variables case too.
We shall denote by $\mathcal{H}(D)$, the class of holomorphic functions
$f:D\rightarrow\mathbb{C}$, where $D\subset\mathbb{C}^{n}$ is a domain, and
the open unit ball in $\mathbb{C}^{n}$ shall be denoted by
$\mathbb{D}^{n}:=\\{z\in\mathbb{C}^{n}:\|z\|<1\\}.$
For every $\psi\in\mathcal{C}^{2}(D),$ at each point $z\in D$ we define a
Hermitian form

$L_{z}(\psi,~{}v):=\sum\limits_{k,l=1}^{n}\frac{\partial^{2}(\psi)}{\partial
z_{k}\partial\bar{z_{l}}}(z)v_{k}\bar{v_{l}}$ (1.1)

and is called the Levi form  of the function $\psi$ at $z.$
The spherical derivative of $f\in\mathcal{H}(D)$ (see Dov1 ) is defined as

$f^{\\#}(z):=\sup\limits_{\|v\|=1}\sqrt{L_{z}(\log(1+\lvert
f\rvert^{2}),~{}v)}.$ (1.2)

Since $L_{z}(\log(1+\lvert f\rvert^{2}),~{}v)\geq 0,\ f^{\\#}(z)$ given by
(1.2) is well defined and for $n=1$, $\eqref{eqB}$ reduces to

$f^{\\#}(z):=\frac{\lvert f^{\prime}(z)\rvert}{1+\lvert f(z)\rvert^{2}}$

which is the spherical derivative of a holomorphic function of one complex
variable. Thus (1.2) gives the natural extension of the spherical derivative
to $\mathbb{C}^{n}.$
Further, from (1.2), one easily finds that

$f^{\sharp}(z)=\frac{\sup\limits_{\|v\|=1}\lvert(Df(z),v)\rvert}{1+\lvert
f(z)\rvert^{2}},$ (1.3)

where

$(Df(z),v)=\sum_{j=1}^{n}\frac{\partial f(z)}{\partial z_{j}}.v_{j}$

Following known results shall be required to accomplish the proofs of main
results of this paper, spread over various sections:

Marty’s Theorem in $\mathbb{C}^{n}$ (Dov1 ): A family
$\mathcal{F}\subset\mathcal{H}(D)$ is normal in $D$ if and only if for each
compact set $K\subset D$ there exists a constant $M=M(K)$ such that
$f^{\\#}(z)\leq M$ for all $f\in\mathcal{F},\ z\in K.$

Zalcman’s Lemma in $\mathbb{C}^{n}$ (Dov2 ): A family
$\mathcal{F}\subseteq\mathcal{H}(D)$ is not normal at a point $z_{0}\in D$ if
and only if for each $\alpha\in(-1,\infty)$ there exist sequences
$\\{f_{j}\\}\subset\mathcal{F},~{}\\{z_{j}\\}\subset D:z_{j}\to z_{0},\mbox{
and }~{}\\{r_{j}\\}\subset(0,1]:r_{j}\to 0$ such that the zoomed sequence

$g_{j}(z):=r_{j}^{\alpha}f_{j}(z_{j}+r_{j}z)$

converges locally uniformly to a non-constant entire function $g$ in
$\mathbb{C}^{n}$ satisfying $g^{\sharp}(z)\leq g^{\sharp}(0)=1.$

######  Definition 1.

Let $f\in\mathcal{H}(\mathbb{C}^{n})$ and
$z=(z_{1},z_{2},\ldots,z_{n})\in\mathbb{C}^{n}.$ The total derivative of $f$,
which we shall denote by $D_{0}f,$ is defined as

$D_{0}f(z)=\displaystyle\sum_{j=1}^{n}z_{j}f_{z_{j}},$

where $f_{z_{j}}$ is the partial derivative of $f$ with respect to $z_{j}.$

Liu and Cao Liu  obtained an extension of Zalcman’s lemma concerning the total
derivative in several complex variables as follows:

######  Lemma 1.

Let $\mathcal{F}\subset\mathcal{H}(\mathbb{D}^{n})$ and suppose that $f(z)\neq
0$ for every $f\in\mathcal{F}$ and for all $z\in\Delta_{\delta}(0)$ for some
$\delta\in(0,1)$ and that $f(z)=0\implies\lvert D_{0}f(z)\rvert\leq A$ for
some $A>0.$ If $\mathcal{F}$ is not normal in $\mathbb{D}^{n},$ then for all
$k\in(-1,1],$ there exist a real number $r\in(0,1],$ and sequences
$\\{z_{j}\\}\subseteq\mathbb{D}^{n}:0<\|z_{j}\|<r,\
\\{f_{j}\\}\subseteq\mathcal{F},$ and $\\{\rho_{j}\\}\subset(0,1]:\rho_{j}\to
0$ such that

$g_{j}(\zeta)=\rho_{j}^{-k}f_{j}(z_{j}e^{\rho_{j}\zeta}),~{}(\zeta\in\mathbb{C})$

converges locally uniformly to a non-constant entire function $g$ in
$\mathbb{C}$ satisfying $g^{\sharp}(\zeta)\leq g^{\sharp}(0)=A+1.$

######  Definition 2.

For $f\in\mathcal{H}(\mathbb{C}^{n})~{}\mbox{and}~{}a\in\mathbb{C}^{n},$ we
can write $f(z)=\sum\limits_{i=0}^{\infty}P_{i}(z-a),$ where $P_{i}(z)$ is
either a homogeneous polynomial of degree $i$ or identically zero. The zero
multiplicity of $f$ at $a$ is defined as $V_{f}(a)=min\\{i:P_{i}\neq 0\\}.$

We have a kind of variant of Lemma 1:

######  Lemma 2.

Let $\mathcal{F}\subset\mathcal{H}(\mathbb{D}^{n}).$ For $f\in\mathcal{F},$
suppose that for each $a\in f^{-1}(\\{b\\})\cap\mathbb{D}^{n},$ zero
multiplicity of $f~{}\mbox{at}~{}a$ is at least $k.$ Then if $\mathcal{F}$ is
not normal, then for each $\alpha\in(0,k),$ there exist a real number
$r\in(0,1)$ and sequences $\\{z_{j}\\}\subset\mathbb{D}^{n}:\|z_{j}\|<r,\
\\{f_{j}\\}\subset\mathcal{F},$ and $\\{\rho_{j}\\}\subset(0,1]:\rho_{j}\to 0$
such that

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

converges locally uniformly to a non-constant entire function $g$ in
$\mathbb{C}^{n}$ satisfying $g^{\sharp}(\zeta)\leq g^{\sharp}(0)=1.$

The proof of Lemma 2 is exactly on the lines of proof of Theorem $5$ in
Charak4  with Lemma $1$ there replaced by

######  Lemma 3.

Let $f\in\mathcal{H}(\mathbb{D}^{n}),$ for each $a\in
f^{-1}(\\{b\\})\cap\mathbb{D}^{n},$ zero multiplicity of $f~{}\mbox{at}~{}a$
be at least $k,$ and $\alpha\in(-1,k).$ Let
$\Omega:=\\{z\in\mathbb{C}^{n}:\|z\|<r<1\\}\times(0,1]$ and define a function
$F:\Omega\rightarrow\mathbb{R}$ as

$F(z,t)=\frac{\left(1-\frac{\|z\|^{2}}{r^{2}}\right)^{1+\alpha}t^{1+\alpha}f^{\\#}(z)\left(1+\lvert
f(z)\rvert^{2}\right)}{\left(1-\frac{\|z\|^{2}}{r^{2}}\right)^{2\alpha}t^{2\alpha}+\lvert
f(z)\rvert^{2}}.$

If $F(z,1)>1$ for some $z:\|z\|<r$, then there exist
$z_{0}\in\mathbb{C}^{n}:\|z_{0}\|<r<1$ and $t\in(0,1)$ such that

$\sup\limits_{\|z\|<r}F(z,t_{0})=F(z_{0},t_{0})=1.$

Since the proof of Lemma 3 is a modification of the proof of Lemma $1$ in
Charak4 , here we only need to establish that

$\lim\limits_{\left(1-\frac{\|z\|^{2}}{r^{2}}\right)t\to 0}F(z,t)=0,$ (1.4)

as a significant amount of modifications is required to establish this limit.

We may assume that $z_{j}\to z_{0},$ where $\|z_{0}\|\leq r.$
If $f(z_{0})\neq 0,$ then

$\lim\limits_{j\to\infty}F(z_{j},t_{j})\leq 0.$

Suppose $f(z_{0})=0.$ Then $f(z)=\sum_{m=s}^{\infty}P_{m}(z-z_{0}),~{}s\geq
k,$ where $P_{m}(z)$ is either a homogeneous polynomial of degree $m$ or
identically zero.

$\displaystyle\lvert f(z_{j})\rvert$ $\displaystyle=$ $\displaystyle\lvert
p_{s}(z_{j}-z_{0})+p_{s+1}(z_{j}-z_{0})+\ldots\rvert$ $\displaystyle=$
$\displaystyle\|z_{j}-z_{0}\|^{s}\left[\frac{\lvert
p_{s}(z_{j}-z_{0})+p_{s+1}(z_{j}-z_{0})+\ldots\rvert}{\|z_{j}-z_{0}\|^{s}}\right]$
$\displaystyle=$ $\displaystyle\lvert a_{s}\rvert\|z_{j}-z_{0}\|^{s}[1+o(1)].$

Similarly, we find that

$\left\lvert\frac{\partial}{\partial z_{i}}f(z_{j})\right\rvert=\lvert
b_{s_{i}}\rvert\|z_{j}-z_{0}\|^{s-1}[1+o(1)].$

We may assume that

$\lim\limits_{j\to\infty}\frac{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)t_{j}}{\|z_{j}-z_{0}\|}=B$

exists. If $B\geq 1,$ then

$\displaystyle\lim\limits_{j\to\infty}F(z_{j},t_{j})$ $\displaystyle\leq$
$\displaystyle\lim\limits_{j\to\infty}\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{1-\alpha}t_{j}^{1-\alpha}f^{\\#}(z_{j})\left(1+\lvert
f(z_{j})\rvert^{2}\right)$ $\displaystyle\leq$
$\displaystyle\frac{\|\left(\frac{\partial}{\partial
z_{1}}f(z_{j}),\ldots,\frac{\partial}{\partial
z_{n}}f(z_{j})\right)\|}{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{\alpha-1}t_{j}^{\alpha-1}}$
$\displaystyle\leq$
$\displaystyle\lim\limits_{j\to\infty}\frac{\lvert\frac{\partial}{\partial
z_{1}}f(z_{j})\rvert+\ldots+\lvert\frac{\partial}{\partial
z_{n}}f(z_{j})\rvert}{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{\alpha-1}t_{j}^{\alpha-1}}$
$\displaystyle\leq$ $\displaystyle\lim\limits_{j\to\infty}\frac{\lvert
b_{s_{1}}\rvert\|z_{j}-z_{0}\|^{s-1}+\ldots\lvert
b_{s_{n}}\rvert\|z_{j}-z_{0}\|^{s-1}}{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{\alpha-1}t_{j}^{\alpha-1}}$
$\displaystyle\leq$
$\displaystyle\lim\limits_{j\to\infty}\frac{n\max_{i=1}^{n}\lvert
b_{s_{i}}\rvert\|z_{j}-z_{0}\|^{s-1}}{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{\alpha-1}t_{j}^{\alpha-1}}$
$\displaystyle=$
$\displaystyle\lim\limits_{j\to\infty}\frac{n\max_{i=1}^{n}\lvert
b_{s_{i}}\rvert\|z_{j}-z_{0}\|^{s-\alpha}\|z_{j}-z_{0}\|^{\alpha-1}}{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{\alpha-1}t_{j}^{\alpha-1}}$
$\displaystyle=$ $\displaystyle 0$

If $B\leq 1,$ then

$F(z_{j},t_{j})=\frac{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{\alpha}t_{j}^{\alpha}\lvert
f(z_{j})\rvert}{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{2\alpha}t_{j}^{2\alpha}+\lvert
f(z_{j})\rvert^{2}}.\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)t_{j}\frac{f^{\\#}(z_{j})\left(1+\lvert
f(z_{j})\rvert^{2}\right)}{\lvert f(z_{j})\rvert}.$

The first factor on the right hand is obviously bounded by $\frac{1}{2}.$
Moreover, if $B\neq 0,$ then we have

$\displaystyle\lim\limits_{j\to\infty}\frac{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{\alpha}t_{j}^{\alpha}\lvert
f(z_{j})\rvert}{\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)^{2\alpha}t_{j}^{2\alpha}+\lvert
f(z_{j})\rvert^{2}}$ $\displaystyle\leq$
$\displaystyle\lim\limits_{j\to\infty}\frac{B^{\alpha}\|z_{j}-z_{0}\|^{\alpha}\lvert
a_{s}\rvert\|z_{j}-z_{0}\|^{s}}{B^{2\alpha}\|z_{j}-z_{0}\|^{2\alpha}}$
$\displaystyle=$ $\displaystyle\lim\limits_{j\to\infty}\frac{\lvert
a_{s}\rvert\|z_{j}-z_{0}\|^{s-\alpha}}{B^{\alpha}}$ $\displaystyle=$
$\displaystyle 0$

On the other hand,

$\displaystyle\lim\limits_{j\to\infty}\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)t_{j}\frac{f^{\\#}(z_{j})\left(1+\lvert
f(z_{j})\rvert^{2}\right)}{\lvert f(z_{j})\rvert}$ $\displaystyle\leq$
$\displaystyle\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)t_{j}.\frac{n\max_{i=1}^{n}\lvert
b_{s_{i}}\rvert\|z_{j}-z_{0}\|^{s-1}}{\lvert a_{s}\rvert\|z_{j}-z_{0}\|^{s}}$
$\displaystyle=$
$\displaystyle\lim\limits_{j\to\infty}\left(1-\frac{\|z_{j}\|^{2}}{r^{2}}\right)t_{j}.\frac{n\max_{i=1}^{n}\lvert
b_{s_{i}}\rvert}{\lvert a_{s}\rvert\|z_{j}-z_{0}\|}$ $\displaystyle=$
$\displaystyle B\frac{n\max_{i=1}^{n}\lvert b_{s_{i}}\rvert}{\lvert
a_{s}\rvert}$

It follows that $\lim\limits_{j\to\infty}F(z_{j},t_{j})=0$.

Let $f$ be meromorphic function in $\mathbb{C}$ and $a\in\mathbb{C}_{\infty}.$
Then $a$ is called totally ramified value of $f$ if $f-a$ has no simple zeros.
Following result known as Nevanlinna’s Theorem  (see Bergweiler ) plays a
crucial role in the proofs of Theorem 7, Theorem 13 and Theorem 14 to follow:

######  Theorem 4.

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

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

##  2 Counterexamples to Bloch’s Principle and its Converse

###  2.1 A Counterexample to Bloch’s Principle

If $D\subset\mathbb{C}^{n}$ and $f\in\mathcal{H}(D)$ satisfies a certain
property $\bf P$ in $D,$ then we write it as $(f,D)\in\bf P.$ Recall that
Bloch’s principle states that a family
$\mathcal{F}\subset\mathcal{H}(D):(f,D)\in\bf P$ is likely to be normal in $D$
if $(f,\mathbb{C}^{n})\in\bf P$ implies that $f$ is a constant function. As in
the case of one complex variable, Bloch’s principle fails to hold in
$\mathbb{C}^{n}$ as well.
Let $\bf P$ be the property of holomorphic function $f$ in $D$ defined as

$P(f)(z)=\displaystyle\prod_{i=1}^{n}\left(\frac{\partial f(z)}{\partial
z_{i}}-a\right)\left(\frac{\partial f(z)}{\partial
z_{i}}-b\right)\left(\frac{\partial f(z)}{\partial z_{i}}-f(z)\right)$

omits the value zero.

######  Proposition 5.

If $f\in\mathcal{H}(\mathbb{C}^{n}):(f,\mathbb{C}^{n})\in\bf P,$ then
$f\equiv$constant.

Proof: Suppose $f\in\mathcal{H}(\mathbb{C}^{n})$ possesses the property $\bf
P$. Let

$w=(a_{1},\ldots,a_{n}),~{}w^{\prime}=(b_{1},\ldots,b_{n})\in\mathbb{C}^{n}$

and define

$h_{i}(z_{i})=f(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n}),~{}i=1,2,\ldots
n.$

Then $\frac{d}{dz_{i}}h_{i}(z_{i})$ omits the values $a$ and $b$ and so
$\frac{d}{dz_{i}}h_{i}(z_{i})=c_{i},$ a constant. This implies that
$h_{i}(z_{i})=c_{i}z_{i}+d_{i},$ but
$\frac{d}{dz_{i}}h_{i}(z_{i})-h_{i}(z_{i})$ omits zero and so
$c_{i}-c_{i}z_{i}-d_{i}\neq 0$ which implies $c_{i}=0.$ This shows that each
$h_{i}(z_{i})$ is constant and hence $f$ is constant.$\Box$

######  Counterexample 6.

Following counterexample to the Bloch’s principle is constructed from the
ideas of RubelRubel .

Take $r$ sufficiently small so that

$\\{(z_{1},\ldots,z_{n}):z_{1}+\ldots+z_{n}=1\\}\cap D^{n}(0,r)=\emptyset.$

Consider

$\mathcal{F}=\\{f_{m}(z_{1},\ldots,z_{n})=m(z_{1}+\ldots+z_{n}):(z_{1},\ldots,z_{n})\in
D^{n}(0,r),m\geq 3\\}.$

Then

$\frac{\partial f(z)}{\partial z_{i}}=m\neq 1,2.$

Also,

$\frac{\partial f(z)}{\partial z_{i}}-f(z)=m(1-(z_{1}+\ldots+z_{n}))\neq
0~{}~{}\forall z=(z_{1},\ldots,z_{n})\in D^{n}(0,r).$

Thus each $f_{m}$ satisfies the property $\bf P$ in $D^{n}(0,r)$ with
$a=1~{}\mbox{and}~{}b=2$ but $\mathcal{F}$ is not normal in $D^{n}(0,r)$
showing that Bloch’s principle does not hold in general.

###  2.2 Counterexamples to the Converse of Bloch’s Principle

The converse of Bloch’s principle precisely states that if

$\mathcal{F}:=\\{f\in\mathcal{H}(D):(f,D)\in{\bf P}\\}$

is normal in $D,$ then any
$f\in\mathcal{H}(\mathbb{C}^{n}):(f,\mathbb{C}^{n})\in{\bf P}$ reduces to a
constant.
Let the property P satisfied by a function $f\in\mathcal{H}(D)$ be defined as
follows:

  * (i)

there exist three distinct complex numbers $a_{1},a_{2},a_{3}$ and three
complex numbers $b_{1},b_{2},b_{3}$ satisfying $f(z)=a_{i}\implies
D_{0}f(z)=b_{i}\mathchar 24635\relax\;$

  * (ii)

$f(z)\neq 0~{}\forall~{}z\in\Delta_{\delta}(0)~{}\mbox{for some}~{}\delta>0.$

This property leads to the following normality criterion:

######  Theorem 7.

A subfamily $\mathcal{F}$ of $\mathcal{H}(D)$ is normal in $D$ if
$(f,D)\in{\bf P}$ for all $f\in\mathcal{F}.$

Proof: Since normality is a local property, we may assume $D$ to be the open
unit ball $\mathbb{D}^{n}:=\\{z\in\mathbb{C}^{n}:\|z\|<1\\}.$ Suppose
$\mathcal{F}$ is not normal in $\mathbb{D}^{n}.$ Then by Lemma 1, there exist
$r\in(0,1),~{}z_{j}\in\mathbb{D}^{n}:0<\|z_{j}\|<r,~{}\\{f_{j}\\}\subseteq\mathcal{F},~{}\\{\rho_{j}\\}\subset(0,1]:\rho_{j}\to
0$ such that

$g_{j}(\zeta)=f_{j}(z_{j}e^{\rho_{j}\zeta})$

converges locally uniformly to non-constant entire function $g$ in
$\mathbb{C}.$ Let $g(\zeta_{0})=a_{i}.$ Then by Hurwitz’s theorem there exists
a sequence $\\{\zeta_{j}\\}$ converging to $\zeta_{0}$ such that for
sufficiently large $j,~{}g_{j}(\zeta_{j})=a_{i}$ which implies
$f_{j}(z_{j}e^{\rho_{j}\zeta_{j}})=a_{i}$ and so by given hypothesis
$D_{0}f_{j}(z_{j}e^{\rho_{j}\zeta_{j}})=b_{i}.$ Therefore,

$g^{\prime}(\zeta_{0})=\lim_{j\to\infty}g_{j}^{\prime}(\zeta_{j})=\lim_{j\to\infty}\rho_{j}D_{0}f_{j}(z_{j}e^{\rho_{j}\zeta_{j}})=0.$

Thus each $a_{i}$-points of $g$ has multiplicity at least two which
contradicts Theorem 4. $\Box$

Theorem 7 leads to a counterexample to the converse of Bloch’s principle.

######  Counterexample 8.

Consider the function

$f(z_{1},z_{2},\ldots,z_{n})=z_{1}+z_{2}+\ldots+z_{n}+4,$

$a_{1}=1,\ a_{2}=2,\ a_{3}=3$ and $b_{1}=-3,\ b_{2}=-2,\ b_{3}=-1.$
Then $f(z)=a_{j}\implies D_{0}f(z)=b_{j}~{}(j=1,2,3).$ Also $f(z)\neq
0~{}\forall~{}z\in\Delta_{\delta}(0)~{}\mbox{for some}~{}\delta>0.$ Thus there
exists a non-constant entire function in $\mathbb{C}^{n}$ satisfying the
property $\bf{P}$ and by Theorem 7,
$\mathcal{F}:=\\{f\in\mathcal{H}(D):(f,D)\in\bf{P}\\}$ is normal in $D$.

The following weak version of Lappan’s normality criterion in several complex
variables(see Charak4 ,Theorem 1) also leads to a counterexample to the
converse of Bloch’s principle:

######  Theorem 9.

Let $D\subseteq\mathbb{C}^{n}$ and $\mathcal{F}\subset\mathcal{H}(D).$ Let $E$
be a subset of $\mathbb{C}$ containing at least three points such that for
each compact set $K\subset D,$ there exists a positive constant $M=M(K)$ for
which

$\sup\limits_{\left\|v\right\|=1}\lvert(Df(z),v)\rvert\leq M\mbox{ whenever
}f(z)\in E,~{}z\in K,~{}f\in\mathcal{F}.$ (2.1)

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

The proof of Theorem 9 is obtained on the lines of the proof of Theorem $1$ in
Charak4  hence omitted.

######  Counterexample 10.

Let $D\subset\mathbb{C}^{n}$ be a domain. For $f\in\mathcal{H}(D),$
$(f,D)\in{\bf P}$ if there exists a set $E\subset\mathbb{C}$ containing at
least three points such that for each compact subset $K\subset D,$ there
exists a positive constant $M=M(K)$ for which

$\sup\limits_{\left\|v\right\|=1}\lvert(Df(z),v)\rvert\leq M\mbox{ whenever
}f(z)\in E,~{}z\in K.$ (2.2)

Consider the function

$f(z_{1},z_{2},\ldots,z_{n})=z_{1}+z_{2}+\ldots+z_{n}.$

Then

$\sup\limits_{\left\|v\right\|=1}\lvert(Df(z),v)\rvert\leq\sup\limits_{\left\|v\right\|=1}[\lvert
v_{1}\rvert+\lvert v_{2}\rvert+\cdots+\lvert v_{n}\rvert]\leq n.$

Thus $f$ is a non-constant entire function in $\mathbb{C}^{n}$ satisfying the
property ${\bf P}$ in $\mathbb{C}^{n}$ and by Theorem 9,
$\mathcal{F}:=\\{f\in\mathcal{H}(D):(f,D)\in{\bf P}\\}$ is normal in $D$. This
is a another counterexample to the converse of Bloch’s principle in several
complex variables.

##  3 Picard-type Theorems and their Corresponding Normality Criteria

By Bloch’s principle one can expect to have a normality criterion
corresponding to every Picard-type theorem. Here we prove a couple of Picard-
type theorems and obtain the corresponding normality criteria.
In the discussions to follow, we shall denote the operator
$\left(\frac{\partial^{k}}{\partial z_{1}^{k}},\frac{\partial^{k}}{\partial
z_{2}^{k}},\ldots,\frac{\partial^{k}}{\partial z_{n}^{k}}\right)$ by $D^{k}.$

######  Theorem 11.

Let $f\in\mathcal{H}(\mathbb{C}^{n})$ and $k$ be a positive integer. If
$f(z)\neq 0,~{}(D^{k}f(z),v)\neq 1~{}\mbox{for
all}~{}~{}v=(v_{1},...,v_{n}):\|v\|=1,$ then $f$ is constant.

Proof: Suppose $f$ is non-constant. Let

$w=(a_{1},\ldots,a_{n}),~{}w^{\prime}=(b_{1},\ldots,b_{n})\in\mathbb{C}^{n}$

and define

$h_{i}(z_{i}):=f(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n}),~{}i=1,2,\ldots
n.$

By given hypothesis, it follows that $h_{i}(z)\neq 0~{}\mbox{for
all}~{}z\in\mathbb{C}.$

Claim: $\frac{\partial^{k}}{\partial
z_{i}^{k}}f(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n})\not\equiv 1$

Suppose

$\frac{\partial^{k}}{\partial
z_{i}^{k}}f(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n})\equiv 1.$

Then $h_{i}(z)$ is a polynomial of degree $k$ and so $h_{i}(a)=0$ for some
$a\in\mathbb{C},$ a contradiction and this establishes the claim.
Since $h_{i}(z)\neq 0~{}\mbox{for all}~{}z\in\mathbb{C}.$ So,
$\frac{\partial^{k}}{\partial z_{i}^{k}}h_{i}(a)=1~{}\mbox{for
some}~{}a\in\mathbb{C}$ which implies

$\frac{\partial^{k}}{\partial
z_{i}^{k}}f(b_{1},\ldots,b_{i-1},a,a_{i+1},\ldots,a_{n})=1.$

which implies that

$\sum\limits_{j=1}^{n}\frac{\partial^{k}}{\partial
z_{j}^{k}}f(b_{1},\ldots,b_{i-1},a,a_{i+1},\ldots,a_{n})e_{j}=1$

where $e_{j}=\left\\{\begin{array}[]{cc}1,&~{}\mbox{for}~{}j=i\\\
0,&~{}\mbox{for}~{}j\neq i.\end{array}\right.$

That is,

$(D^{k}f(b_{1},\ldots,b_{i-1},a,a_{i+1},\ldots,a_{n}),v)=1,~{}\mbox{where}~{}v=(0,\ldots,1,\ldots,0),$

which is a contradiction. $\Box$

The corresponding normality criterion of Theorem 11 is

######  Theorem 12.

Let $D$ be a domain in $\mathbb{C}^{n}$. Let $\mathcal{F}$ be a family of
holomorphic functions in a domain $D$ and $k$ be a positive integer. If
$f(z)\neq 0,~{}(D^{k}f(z),v)\neq 1$ for all
$f\in\mathcal{F},~{}v=(v_{1},...,v_{n}):\|v\|=1,$ then $\mathcal{F}$ is normal
in $D.$

Proof: Suppose $\mathcal{F}$ is not normal in $D.$ Then by Lemma 2, there
exist sequences $f_{j}\in\mathcal{F},$ $z_{j}\rightarrow z_{0},$
$\rho_{j}\rightarrow 0,$ such that the sequence $\\{g_{j}\\}$ defined as
$g_{j}(\zeta)=\frac{f_{j}(z_{j}+\rho_{j}\zeta)}{\rho_{j}^{k}}$ converges
locally uniformly in $\mathbb{C}^{n}$ to a non-constant entire function $g.$
Let
$w=(a_{1},\ldots,a_{n}),~{}w^{\prime}=(b_{1},\ldots,b_{n})\in\mathbb{C}^{n}$
and define

$h_{i}(z_{i}):=g(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n}),~{}i=1,2,\ldots
n.$

By Hurwitz’s theorem, it follows that $h_{i}(z)\neq 0~{}\mbox{for
all}~{}z\in\mathbb{C}.$
By Weierstrass’s theorem, we have

$\frac{\partial^{k}}{\partial
z_{i}^{k}}g_{j}(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n})\to\frac{\partial^{k}}{\partial
z_{i}^{k}}g(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n}).$

Claim: $\frac{\partial^{k}}{\partial
z_{i}^{k}}g(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n})\not\equiv 1$

Suppose $\frac{\partial^{k}}{\partial
z_{i}^{k}}g(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n})\equiv 1.$ Then
$h_{i}(z)$ is a polynomial of degree $k$ and so $h_{i}(a)=0$ for some
$a\in\mathbb{C},$ a contradiction and this establishes the claim.

Since $h_{i}(z)\neq 0~{}\mbox{for all}~{}z\in\mathbb{C}.$ So,
$\frac{\partial^{k}}{\partial z_{i}^{k}}h_{i}(a)=1~{}\mbox{for
some}~{}a\in\mathbb{C}$ which implies

$\frac{\partial^{k}}{\partial
z_{i}^{k}}g(b_{1},\ldots,b_{i-1},a,a_{i+1},\ldots,a_{n})=1.$

Since $\frac{\partial^{k}}{\partial z_{i}^{k}}g$ is non-constant. So, by
Hurwitz’s theorem there is a sequence
$w_{j}=(b_{1},\ldots,b_{i-1},a_{j},a_{i+1},\ldots,a_{n})\to
w_{0}=(b_{1},\ldots,b_{i-1},a,a_{i+1},\ldots,a_{n})$ such that for
sufficiently large $j$

$\frac{\partial^{k}}{\partial z_{i}^{k}}g_{j}(w_{j})=1$

which implies

$\sum\limits_{l=1}^{n}\frac{\partial^{k}}{\partial
z_{l}^{k}}f_{j}(z_{j}+\rho_{j}w_{j})e_{l}=1$

where $e_{l}=\left\\{\begin{array}[]{cc}1,&~{}\mbox{for}~{}l=i\\\
0,&~{}\mbox{for}~{}l\neq i.\end{array}\right.$
That is,
$(D^{k}f_{j}(z_{j}+\rho_{j}w_{j}),v)=1,~{}\mbox{where}~{}v=(0,\ldots,1,\ldots,0),$
which is a contradiction and hence $\mathcal{F}$ is normal in $D.$ $\Box$

The next Picard-type theorem is

######  Theorem 13.

Let $f\in\mathcal{H}(\mathbb{C}^{n})~{}\mbox{and}~{}\ a~{}\mbox{and}~{}b$ be
two distinct finite complex numbers. If $f\neq a$ and for each $\alpha\in
f^{-1}(\\{b\\}),$ zero multiplicity of $f$ at $\alpha$ is at least two, then
$f$ is constant.

Proof: Let
$w=(a_{1},\ldots,a_{n}),~{}w^{\prime}=(b_{1},\ldots,b_{n})\in\mathbb{C}^{n}$
and define

$h_{i}(z_{i})=f(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n}),~{}i=1,2,\ldots
n.$

By the given hypothesis $h_{i}$ omits $a.$ Suppose $h_{i}(\alpha_{i})=b.$ Then

$f(b_{1},\ldots,b_{i-1},\alpha_{i},a_{i+1},\ldots,a_{n})=b$

and so $B_{i}=(b_{1},\ldots,b_{i-1},\alpha_{i},a_{i+1},\ldots,a_{n})\in
f^{-1}(\\{b\\}).$ Since for each $\alpha\in f^{-1}(\\{b\\}),$ zero
multiplicity of $f$ at $\alpha$ is at least two, we have

$f(z)=\sum\limits_{m=2}^{\infty}P_{m}(z-B_{i}),$

where $P_{m}$ is either identically zero or a homogeneous polynomial of degree
$m.$
Now

$\frac{\partial}{\partial z_{i}}f(z)=\sum\limits_{m=1}^{\infty}q_{m}(z-B_{i})$

where $q_{m}$ is either identically zero or a homogeneous polynomial of degree
$m.$
Thus $\frac{\partial}{\partial z_{i}}f(B_{i})=0$ which implies
$\frac{d}{dz_{i}}h_{i}(\alpha_{i})=0.$ Thus $b-$points of $h_{i}$ have
multiplicity at least two and so by Theorem 4, we conclude that each
$h_{i}(z_{i})$ is constant and hence $f$ is constant. $\Box$

The corresponding normality criterion of Theorem 13 is

######  Theorem 14.

Let $D$ be a domain in $\mathbb{C}^{n}$. Let
$\mathcal{F}\subset\mathcal{H}(D),\ a~{}\mbox{and}~{}b$ be two distinct finite
complex numbers. If for each $f\in\mathcal{F},~{}f(z)\neq a$ and for each
$\alpha\in f^{-1}(\\{b\\})\cap D,$ zero multiplicity of $f$ at $\alpha$ is at
least two, then $\mathcal{F}$ is normal in $D.$

Proof: Suppose $\mathcal{F}$ is not normal in $D.$ Then by Lemma 2, there
exist sequences $f_{j}\in\mathcal{F},$ $z_{j}\rightarrow z_{0},$
$\rho_{j}\rightarrow 0,$ such that the sequence $\\{g_{j}\\}$ defined as
$g_{j}(\zeta)=f_{j}(z_{j}+\rho_{j}\zeta)$ converges locally uniformly in
$\mathbb{C}^{n}$ to a non-constant entire function $g.$ Let
$w=(a_{1},\ldots,a_{n}),~{}w^{\prime}=(b_{1},\ldots,b_{n})\in\mathbb{C}^{n}$
and define

$h_{i}(z_{i})=g(b_{1},\ldots,b_{i-1},z_{i},a_{i+1},\ldots,a_{n}),~{}i=1,2,\ldots
n.$

By Hurwitz’s theorem, it follows that $h_{i}(z)\neq 0~{}\mbox{for
all}~{}z\in\mathbb{C}.$
Suppose $h_{i}(\alpha_{i})=b$ which implies
$g(b_{1},\ldots,b_{i-1},\alpha_{i},a_{i+1},\ldots,a_{n})=b.$ By Hurwitz’s
theorem there exist sequences
$\beta_{j,i}=(b_{1},\ldots,b_{i-1},\alpha_{j,i},a_{i+1},\ldots,a_{n})\to\beta=(b_{1},\ldots,b_{i-1},\alpha_{i},a_{i+1},\ldots,a_{n})$
such that $f_{j}(z_{j}+\rho_{j}\beta_{j,i})=b$ which implies
$z_{j}+\rho_{j}\beta_{j,i}\in f_{j}^{-1}(\\{b\\})\cap D,$ for sufficiently
large $j.$ Since for $\alpha\in f_{j}^{-1}(\\{b\\})\cap D,$ zero multiplicity
of $f_{j}$ at $\alpha$ is at least two, we have
$f_{j}(z)=\sum\limits_{m=2}^{\infty}P_{j_{m}}(z-(z_{j}+\rho_{j}\beta_{j,i})),$
where $P_{j_{m}}$ is either identically zero or a homogeneous polynomial of
degree $m.$
Thus $\frac{\partial}{\partial
z_{i}}(f_{j}(z))=\sum\limits_{m=1}^{\infty}q_{j_{m}}(z-(z_{j}+\rho_{j}\beta_{j,i})),$
where $q_{j_{m}}$ is either identically zero or a homogeneous polynomial of
degree $m.$
Thus

$\frac{\partial}{\partial z_{i}}(f_{j}(z_{j}+\rho_{j}\beta_{j,i}))=0$

which implies

$\frac{\partial}{\partial
z_{i}}g_{j}(b_{1},\ldots,b_{i-1},\alpha_{j,i},a_{i+1},\ldots,a_{n})=0$

Taking $j\to\infty,~{}\frac{\partial}{\partial
z_{i}}g(b_{1},\ldots,b_{i-1},\alpha_{i},a_{i+1},\ldots,a_{n})=0$ which implies
$\frac{d}{dz_{i}}h_{i}(\alpha_{i})=0.$ Thus $b-$ points of $h_{i}$ have
multiplicity atleast two and so by Theorem 4 each $h_{i}(z_{i})$ is constant
and hence $g$ is constant, a contradiction. $\Box$

The fundamental normality test for holomorphic functions of several complex
variables is a simple consequence of Theorem 14.

##  4 Extension of Some Known Results from $\mathbb{C}$ to $\mathbb{C}^{n}$

In $2012$, J. Grahl and S. NevoNevo  proved the following normality criterion:

######  Theorem 15.

Let some $\epsilon>0$ be given and set
$\mathcal{F}:=\\{f\in\mathcal{H}(\Omega):f^{\sharp}(z)\geq\epsilon~{}\mbox{for
all}~{}z\in\Omega\\}.$ Then $\mathcal{F}$ is normal in $\Omega.$

We find a several complex variables analogue of Theorem 15:

######  Theorem 16.

Let $D$ be a domain in $\mathbb{C}^{n}$. Let $k\geq 0$ be an integer,
$c>0,~{}\alpha>1$ constant. Then

$\mathcal{F}=\left\\{f\in\mathcal{H}(D):\sup\limits_{\|v\|=1}\frac{\lvert(D^{k}f(z),v)\rvert}{1+\lvert
f(z)\rvert^{\alpha}}>c\right\\}$

is normal in $D.$

Proof: Suppose $\mathcal{F}$ is not normal in $D.$ Taking
$\beta>\frac{k}{\alpha-1},$ by Zalcman’s lemma in $\mathbb{C}^{n},$ there
exist sequences $f_{j}\in\mathcal{F},$ $z_{j}\rightarrow z_{0},$
$\rho_{j}\rightarrow 0,$ such that the sequence $\\{g_{j}\\}$ defined as

$g_{j}(\zeta)=\rho_{j}^{\beta}f_{j}(z_{j}+\rho_{j}\zeta)$

converges locally uniformly in $\mathbb{C}^{n}$ to a non-constant entire
function $g.$ Let $\zeta_{0}\in\mathbb{C}^{n}$ such that $g(\zeta_{0})\neq 0.$
Clearly $f_{j}(z_{j}+\rho_{j}\zeta_{0})\to\infty~{}\mbox{as}~{}j\to\infty.$
Also

$\sup\limits_{\|v\|=1}\lvert(D^{k}g_{j}(\zeta_{0}),v)\rvert=\sup\limits_{\|v\|=1}\rho_{j}^{\beta+k}\lvert(D^{k}f_{j}(z_{j}+\rho_{j}\zeta_{0}),v)\rvert\to\sup\limits_{\|v\|=1}\lvert(D^{k}g(\zeta_{0}),v)\rvert.$

Since

$\sup\limits_{\|v\|=1}\lvert(D^{k}f_{j}(z_{j}+\rho_{j}\zeta_{0}),v)\rvert>c\lvert
f_{j}(z_{j}+\rho_{j}\zeta_{0})\rvert^{\alpha},$

we have

$\displaystyle\sup\limits_{\|v\|=1}\lvert(D^{k}g_{j}(\zeta_{0}),v)\rvert$
$\displaystyle=$
$\displaystyle\sup\limits_{\|v\|=1}\rho_{j}^{\beta+k}\lvert(D^{k}f_{j}(z_{j}+\rho_{j}\zeta_{0}),v)\rvert$
$\displaystyle>$ $\displaystyle\rho_{j}^{\beta+k}c\lvert
f_{j}(z_{j}+\rho_{j}\zeta_{0})\rvert^{\alpha}$ $\displaystyle=$ $\displaystyle
c(\rho_{j}^{\beta}\lvert
f_{j}(z_{j}+\rho_{j}\zeta_{0})\rvert)^{\alpha}.\rho_{j}^{\beta+k-\beta\alpha}\to\infty~{}\mbox{as}~{}j\to\infty.$

which is a contradiction to the fact that $g$ is holomorphic at $\zeta_{0}.$
Therefore, $\mathcal{F}$ has to be normal. $\Box$

######  Remark 1.

For $k=1~{}\mbox{and}~{}\alpha=2,$ we obtain several complex variable version
of Theorem 15.

Further, we obtain an improvement of a Marty’s Theorem in $\mathbb{C}^{n}$:

######  Theorem 17.

Let $D$ be a domain in $\mathbb{C}^{n},$ $\mathcal{F}\subset\mathcal{H}(D)$
and $\alpha$ be a positive real number. If

$F_{\alpha}=\left\\{\sup\limits_{\|v\|=1}\frac{\lvert(Df(z),v)\rvert}{1+\lvert
f(z)\rvert^{\alpha}}:f\in\mathcal{F}\right\\}$

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

Proof: Suppose $\mathcal{F}$ is not normal in $D.$ Then by Lemma 2, there
exist sequences $f_{j}\in\mathcal{F},$ $z_{j}\rightarrow z_{0},$
$\rho_{j}\rightarrow 0,$ such that the sequence $\\{g_{j}\\}$ defined as
$g_{j}(\zeta)=\rho_{j}^{-1/2}f_{j}(z_{j}+\rho_{j}\zeta)$ converges locally
uniformly in $\mathbb{C}^{n}$ to a non-constant entire function $g.$
By Weierstrass’ Theorem,

$\frac{\partial}{\partial z_{i}}g_{j}(\zeta)\to\frac{\partial}{\partial
z_{i}}g(\zeta),~{}i=1,2,\ldots,n.$

Now

$\displaystyle\sup\limits_{\|v\|=1}\sum_{l=1}^{n}\left\lvert\frac{\partial}{\partial
z_{l}}g_{j}(\zeta).v_{l}\right\rvert$ $\displaystyle=$
$\displaystyle\rho_{j}^{1/2}\sup\limits_{\|v\|=1}\sum_{l=1}^{n}\left\lvert\frac{\partial}{\partial
z_{l}}f_{j}(z_{j}+\rho_{j}\zeta).v_{l}\right\rvert$ $\displaystyle=$
$\displaystyle\rho_{j}^{1/2}\sup\limits_{\|v\|=1}\lvert(Df_{j}(z_{j}+\rho_{j}\zeta),v)\rvert$
$\displaystyle\leq$ $\displaystyle\rho_{j}^{1/2}M[1+\lvert
f_{j}(z_{j}+\rho_{j}\zeta)\rvert^{\alpha}]$ $\displaystyle=$
$\displaystyle\rho_{j}^{1/2}M[1+\lvert\rho_{j}^{1/2}g_{j}(\zeta)\rvert^{\alpha}]\to
0~{}\mbox{as}~{}j\to\infty$

Thus

$\sup\limits_{\|v\|=1}\sum_{l=1}^{n}\left\lvert\frac{\partial}{\partial
z_{l}}g_{j}(\zeta).v_{l}\right\rvert=0.$

That is,

$\frac{\partial g}{\partial z_{i}}(\zeta)=0,~{}i=1,\ldots,n,$

which implies $g$ is a constant, a contradiction. $\Box$

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