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# Necessary and sufficient conditions
for zero subsets of holomorphic functions

B. N. Khabibullin Bashkir State University, Zaki Validi Str. 32,
Bashkortostan, 420076 Russian Federation F. B. Khabibullin Ufa State Aviation
Technical University, Karl Marx Str. 12, Bashkortostan, 420008 Russian
Federation

(December 14, 2020)

###### Abstract

Let $D$ be a domain in the complex plane, $M$ be an extended real function on
$D$. If $f$ is a non-zero holomorphic function on $D$ with an upper constraint
$|f|\leq\exp M$ on this domain $D$, then it is natural to expect that there
must be some upper constraints on the distribution of zeros of this
holomorphic function exclusively in terms of the function $M$ and the geometry
of the domain $D$. We have investigated this question in detail in our
previous works in the case when $M$ is a subharmonic function and the domain
$D$ is arbitrary or with a non-polar boundary. The answer was given in terms
of limiting the distribution of zeros of $f$ from above via the Riesz measure
of the subharmonic function $M$. In this article, the function $M$ is the
difference of subharmonic functions, or a $\delta$-subharmonic function, and
the upper constraints are given in terms of the Riesz charge of this
$\delta$-subharmonic function $M$. These results are also new to a certain
extent for the subharmonic function $M$. The case when the domain D is the
complex plane is considered separately. For the complex plane, it is possible
to reach the criterion level.

holomorphic function, zero set, entire function, subharmonic function, Riesz
measure, $\delta$-subharmonic function, Jensen measure

##  1 Introduction

As usual, $\mathbb{N}:=\\{1,2,\dots\\}$, $\mathbb{R}$,
${\mathbb{R}}^{+}:=\bigl{\\{}x\in\mathbb{R}\bigm{|}x\geq 0\bigr{\\}}$,
$\mathbb{C}$ are sets of all natural, real, positive, complex numbers,
respectively. We often denote singleton sets $\\{x\\}$ by its inner symbol $x$
without curly braces. So, $\mathbb{N}_{0}:=0\cup\mathbb{N}$, and
${\mathbb{R}}^{+}\setminus 0$ is the set of all strictly positive numbers,
$\overline{\mathbb{R}}:=-\infty\cup\mathbb{R}\cup+\infty$ is the extended real
line, $\overline{\mathbb{R}}^{+}:=\mathbb{R}^{+}\cup+\infty$, and
$\mathbb{C}_{\infty}:=\mathbb{C}\cup\infty$ is the extended complex plane.
Each this set is endowed with its natural order, algebraic, geometric and
topological structure with modulus $|z|$ and conjugate number $\bar{z}$ for
$z\in\mathbb{C}$, and $\inf\varnothing:=+\infty:=\sup\mathbb{R}$,
$\sup\varnothing:=-\infty:=\inf\mathbb{R}$ for the empty set $\varnothing$.
For a subset $S\subset\mathbb{C}_{\infty}$, $\operatorname{{\sf Hol}}(S)$ is
the algebra over $\mathbb{C}$ of all holomorphic functions $f$ on an open
subset $O_{f}\supset S$ in $\mathbb{C}_{\infty}$. So, $\operatorname{{\sf
Hol}}(\mathbb{C})$ is the algebra of all entire functions.

Let $S$ be a set of points, $J$ be a index set, ${\sf Z}:=\\{{\sf
z}_{j}\\}_{j\in J}$ be indexed set of points ${\sf z}_{j}\in S$. These indexed
sets will be called the distribution of points in $S$. We write $z\in{\sf Z}$
if there is ${\sf z}_{j}=z$, and ${\sf Z}\subset S^{\prime}\subset S$ if ${\sf
z}_{j}\in S^{\prime}$ for each $j\in J$. The counting function $n_{\sf
Z}\colon S\to\overline{\mathbb{N}}$ of $\sf Z$ is defined by

$\displaystyle n_{\sf Z}(z)$ $\displaystyle:=\sum_{{\sf
z}_{j}=z}1\in\overline{\mathbb{N}}_{0}:=\mathbb{N}_{0}\cup+\infty\quad\text{at
each $z\in S$.}$ (1z) We denote by the same symbol $n_{\sf Z}$ the counting
measure $n_{\sf Z}\colon 2^{S}\to\overline{\mathbb{N}}_{0}$ of ${\sf Z}$
defined by $\displaystyle n_{\sf Z}(S^{\prime})$
$\displaystyle\overset{\eqref{df:divz}}{:=}\sum_{{\sf z}_{j}\in
S^{\prime}}1=\sum_{z\in S^{\prime}}n_{\sf
Z}(z)\in\overline{\mathbb{N}}\quad\text{ for each $S^{\prime}\subset S$.}$
(1n)

We say that distributions ${\sf Z}$ and ${\sf Z^{\prime}}$ of points in $S$
are equal and write ${\sf Z}={\sf Z^{\prime}}$ if $n_{\sf Z}=n_{{\sf
Z}^{\prime}}$ on $S$. Thus, both the counting function and the counting
measure uniquely determine distributions of points in $S$. We write ${\sf
Z}\subset{\sf Z}^{\prime}$ if $n_{\sf Z}\leq n_{{\sf Z}^{\prime}}$ on $S$.

Let $O\neq\varnothing$ be an open subset in $\mathbb{C}$. The zero set of
$f\in\operatorname{{\sf Hol}}(O)$ is a distribution $\operatorname{{\sf
Zero}}_{f}$ of points in $O$ with the counting function of the multiplicity of
zeros of $f$ defined by

$n_{\operatorname{{\sf Zero}}_{f}}(z)\underset{\text{\tiny$z\in
O$}}{:=}\sup\left\\{p\in\mathbb{N}_{0}\Bigm{|}\limsup_{z\neq z^{\prime}\to
z}\frac{\bigl{|}f(z)\bigr{|}}{|z^{\prime}-z|^{p}}<+\infty\right\\}.$

So, for the zero function $0\in\operatorname{{\sf Hol}}(O)$, we have
$n_{\operatorname{{\sf Zero}}_{0}}(z)\overset{\eqref{df:divz}}{=}+\infty$ for
each $z\in O$ and $n_{\operatorname{{\sf
Zero}}_{0}}(S)\overset{\eqref{df:divmn}}{=}+\infty$ for each $S\subset O$. The
counting function $n_{\operatorname{{\sf Zero}}_{f}}$ of
$f\in\operatorname{{\sf Hol}}(O)$ is often called the divisor of zeros of $f$.
If ${\sf Z}\subset\operatorname{{\sf Zero}}_{f}$, then we write $f({\sf Z})=0$
and say that ${\sf Z}$ is a zero subset of $f$ and $f$ vanishes on ${\sf Z}$.

For a subset $S\subset\mathbb{C}_{\infty}$ we let $\complement
S:=\mathbb{C}_{\infty}\setminus S$, $\operatorname{{\sf clos}}S$,
$\operatorname{{\sf int}}S:=\complement(\operatorname{{\sf clos}}\complement
S)$, and $\partial S:=\operatorname{{\sf clos}}S\setminus\operatorname{{\sf
int}}S$ denote its complement, closure, interior, and boundary always in
$\mathbb{C}_{\infty}$. We write $S\Subset O$ if $\operatorname{{\sf
clos}}S\subset O$. A distribution of points ${\sf Z}\subset O$ is local finite
if $n_{\sf Z}(S)<+\infty$ for each $S\Subset O$.

######  Weierstrass Theorem.

Let ${\sf Z}$ be a distribution of points in an open set $O\subset\mathbb{C}$.
The following three statements are equivalent:

  * •

There is $f\in\operatorname{{\sf Hol}}(O)$ such that $f\neq 0$ and ${\sf
Z}=\operatorname{{\sf Zero}}_{f}$.

  * •

There is $f\in\operatorname{{\sf Hol}}(O)$ such that $f\neq 0$ and $f({\sf
Z})=0$.

  * •

$\sf Z$ is local finite.

Throughout this article, we consider only locally finite distributions ${\sf
Z}$ of points in an open connected subset $D\subset\mathbb{C}$, i.e. $D$ is a
domain in the complex plane $\mathbb{C}$.

If an additional constraint $\ln|f|\leq M$ on $D$ is imposed on a holomorphic
function $f\neq 0$, where $M\colon D\to\overline{\mathbb{R}}$ is an extended
real function, then the problem of describing zero sets and zero subsets
becomes much more complicated. In particular, zero sets are very often not the
same as zero subsets. In our paper, we consider only zero subsets under a
restriction from above of the form $\ln|f|\leq M$, when $M$ is the difference
of subharmonic functions with the Riesz charge $\varDelta_{M}$. For
subharmonic functions $M$ Rans –Helms , this question was considered earlier
in our series of works KhaRoz18 –MenKha20 . In this article, we consider the
difference $M=M_{\operatorname{\text{\rm\tiny
up}}}-M_{\operatorname{\text{\rm\tiny low}}}$ of subharmonic functions
$M_{\operatorname{\text{\rm\tiny up}}}$ and $M_{\operatorname{\text{\rm\tiny
low}}}$ with the Riesz measures $\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}$ of $M_{\operatorname{\text{\rm\tiny up}}}$ and
$\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}}$ of
$M_{\operatorname{\text{\rm\tiny low}}}$ and with the Riesz charge
$\varDelta_{M}:=\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}-\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}}$ of $M$. It is shown
that if $\ln|f|\leq M$, $f\neq 0$ and $f({\sf Z})=0$, then there is a number
$C\in\mathbb{R}^{+}$ such that

$\sum_{{\sf z}_{j}}v({\sf z}_{j})\leq\int v\operatorname{\,{\rm
d}}\varDelta_{M}+C$ (2)

when $v$ runs through a very wide class of test positive functions. These are
the necessary conditions from Sec. 2, Theorem 1. Conversely, if (2) is
satisfied for much more narrower subclasses of test positive smooth
subharmonic functions $v$, then we have almost converse statements in the form
$\ln|f|\leq M^{\odot}+R$, where $M^{\odot}$ are certain averaging of $M$ over
small disks, and $R$ is a small addition related to the distance to the
boundary $\partial D$ of $D$. These are sufficient conditions from Sec. 3,
Theorem 2. In Sec. 3, we consider domains $D$ with non-polar boundary
$\partial D$, or, equivalently, with non-polar complement $\mathbb{C}\setminus
D$. This very broad class of domains $D$ includes most of the considered in
function theory and its applications. So, if $\mathbb{C}\setminus D$ contains
a connected subset with more than one point, then the boundary $\partial D$ is
non-polar (Rans, , Corollary 3.6.4), (Helms, , Theorem 5.12). But the boundary
$\partial\mathbb{C}=\infty\in\mathbb{C}_{\infty}$ of $\mathbb{C}$ is polar.
For the case of $D=\mathbb{C}$ in Sec. 4, Theorem 3, we obtain a criterion for
a wide class of functions $M$ under a single condition: there are numbers
$P,C\in\mathbb{R}^{+}$ such that

$\int_{0}^{2\pi}M_{\operatorname{\text{\rm\tiny
up}}}\Bigl{(}z+\frac{1}{(1+|z|)^{P}}e^{i\theta}\Bigr{)}\operatorname{\,{\rm
d}}\theta\leq M_{\operatorname{\text{\rm\tiny up}}}(z)+C\quad\text{for each
$z\in\mathbb{C}$.}$ (3)

##  2 Necessary conditions for zero subsets of holomorphic functions with
upper constraints in domain

###  2.1 Subharmonic functions and measures

Let $\operatorname{{\sf sbh}}(S)$ be the cone over $\mathbb{R}^{+}$ of all
subharmonic functions $u$ on open neighborhoods of
$S\subset\mathbb{C}_{\infty}$ including $(-\infty)$-function
$\bm{-\infty}\colon z\underset{z\in S}{\longmapsto}-\infty$, and
$\operatorname{{\sf har}}(S):=\operatorname{{\sf
sbh}}(S)\cap\bigl{(}-\operatorname{{\sf sbh}}(S)\bigr{)}$ be the space over
$\mathbb{R}$ of all harmonic functions $u$ on $S$. We denote by

$\operatorname{{\sf sbh}}_{*}(D):=\bigl{\\{}u\in\operatorname{{\sf
sbh}}(D)\bigm{|}u\neq\bm{-\infty}\bigr{\\}}$

the class of all nontrivial subharmonic functions on a domain
$D\subset\mathbb{C}$.

If $u\in\operatorname{{\sf sbh}}_{*}(D)$, then its Riesz measure is denoted by

$\varDelta_{u}:=\frac{1}{2\pi}{\bigtriangleup}u\in\operatorname{{\sf
Meas}}^{+}(D),$

where ${\bigtriangleup}$ is the Laplace operator acting in the sense of the
theory of distributions or generalized functions, and $\operatorname{{\sf
Meas}}^{+}(D)$ is the cone over $\mathbb{R}^{+}$ of positive Radon measures on
$D$. But by definition, the Riesz measure of $(-\infty)$-function
$\bm{-\infty}$ on $D$ is such that $\varDelta_{\bm{-\infty}}(S)=+\infty$ for
each $S\subset D$.

If $f\in\operatorname{{\sf Hol}}(D)$, then $\ln|f|\in\operatorname{{\sf
sbh}}(D)$ and $n_{\operatorname{{\sf Zero}}_{f}}=\varDelta_{\ln|f|}$ (Rans, ,
3.7.8).

Given $f\in F\subset{\overline{\mathbb{R}}}^{X}$, we set $f^{+}\colon
x\mapsto\max\\{0,f(x)\\}$, and $F^{+}:=\\{f\in F\colon f=f^{+}\\}$. For a
sequence $(f_{k})_{k\in\mathbb{N}}\subset\overline{\mathbb{R}}^{X}$, we write
$f_{k}\underset{k\to\infty}{\nearrow}f$ if this sequence
$(f_{k})_{k\in\mathbb{N}}$ is increasing and
$f=\lim\limits_{k\to\infty}f_{k}$;

$F^{\uparrow}:=\biggl{\\{}f\in\overline{\mathbb{R}}^{X}\Bigm{|}\;\exists(f_{k})\subset
F,\;f_{k}\underset{k\to\infty}{\nearrow}f\biggr{\\}}.$ (4)

So, $\operatorname{{\sf sbh}}^{+}(S):=\bigl{(}\operatorname{{\sf
sbh}}(S)\bigr{)}^{+}$ is the class of all positive subharmonic functions on
$S$. Let $S\Subset D$ be a closed subset. We define classes (KhaRoz18, ,
Definition 1), KhaKha19

$\operatorname{{\sf sbh}}_{0}^{+}(D\setminus S;\leq
b):=\Bigl{\\{}v\in\operatorname{{\sf sbh}}^{+}(D\setminus S)\Bigm{|}v\leq
b\text{ on $D\setminus S$},\quad\limsup_{D\ni z^{\prime}\to
z}v(z^{\prime})\underset{\text{\tiny$z\in\partial D$}}{=}0\Bigr{\\}}.$ (5)

of test subharmonic positive functions $v$ for $D$ outside of $S$ with an
upper bound of $b\in\mathbb{R}^{+}$, and

$\operatorname{{\sf sbh}}_{0}^{+\uparrow}(D\setminus S;\leq
b)\overset{\eqref{Fuparrow}}{:=}\bigl{(}\operatorname{{\sf
sbh}}_{0}^{+}(D\setminus S;\leq b)\bigr{)}^{\uparrow}$

of test upper positive functions $v$ for $D$ outside of $S$ with an upper
bound of $b\in\mathbb{R}^{+}$.

The difference of two nontrivial subharmonic functions is called a nontrivial
$d$-subharmonic function, or a nontrivial $\delta$-subharmonic function. A
$d$-subharmonic function Arsove53 , (KhaRoz18, , 3.1), Az

$M:=M_{\text{\tiny\rm up}}-M_{\text{\tiny\rm low}},\quad M_{\text{\tiny\rm
up}}\in\operatorname{{\sf sbh}}_{*}(D),\quad M_{\text{\tiny\rm
low}}\in\operatorname{{\sf
sbh}}_{*}(D),\quad\varDelta_{M}:=\varDelta_{M_{\text{\tiny\rm
up}}}-\varDelta_{M_{\text{\tiny\rm low}}},$ (6)

with the Riesz charge $\varDelta_{M}$ of $M$ is defined at each point $z\in D$
where $M_{\text{\tiny\rm low}}(z)\neq-\infty$. Below we put $M(z):=+\infty$ if
$M_{\text{\tiny\rm low}}(z)=-\infty$.

For a Borel subset $S\subset\mathbb{C}$, we denote by $\operatorname{{\sf
Meas}}(S)$ the class of all Borel signed measures, or charges, on $S$.
$\operatorname{{\sf Meas}}_{\operatorname{cmp}}(S)$ is the class of charges
$\mu\in\operatorname{{\sf Meas}}(S)$ with a compact support
$\operatorname{supp}\mu\Subset S$;

$\operatorname{{\sf Meas}}^{+}(S):=(\operatorname{{\sf
Meas}}(S))^{+},\quad\operatorname{{\sf
Meas}}_{\operatorname{cmp}}^{+}(S):=(\operatorname{{\sf
Meas}}_{\operatorname{cmp}}(S))^{+}.$

For a charge $\mu\in\operatorname{{\sf Meas}}(S)$, we let $\mu^{+}$,
$\mu^{-}:=(-\mu)^{+}$ and $|\mu|:=\mu^{+}+\mu^{-}$ respectively denote its
upper, lower, and total variations. A function $f\colon
S\to\overline{\mathbb{R}}$ is $\mu$-integrable if there are four integrals
$\int f^{\pm}\operatorname{\,{\rm d}}\mu^{\pm}\in\overline{\mathbb{R}}^{+}$
such that $\int f^{+}\operatorname{\,{\rm d}}\mu^{+}+\int
f^{-}\operatorname{\,{\rm d}}\mu^{-}<+\infty$ for $\int
f^{-}\operatorname{\,{\rm d}}\mu^{+}+\int f^{+}\operatorname{\,{\rm
d}}\mu^{-}=+\infty$, and vice versa, $\int f^{-}\operatorname{\,{\rm
d}}\mu^{+}+\int f^{+}\operatorname{\,{\rm d}}\mu^{-}<+\infty$ for $\int
f^{+}\operatorname{\,{\rm d}}\mu^{+}+\int f^{-}\operatorname{\,{\rm
d}}\mu^{-}=+\infty$. A $\mu$-integrable function $f\colon
S\to\overline{\mathbb{R}}$ is $\mu$-summable if $\int|f|\operatorname{\,{\rm
d}}|\mu|\neq+\infty$.

###  2.2 Jensen measures and its potentials

A measure $\mu\in\operatorname{{\sf Meas}}_{\operatorname{cmp}}^{+}(D)$ is a
Jensen measure for domain $D\subset D$ at $z_{0}\in D$ if Kha03 , KhaRoz18

$u(z_{0})\leq\int u\operatorname{\,{\rm d}}\mu\quad\text{for each
$z_{0}\in\operatorname{{\sf sbh}}(D)$}.$ (7)

We denote by ${\sf J}_{z_{0}}(D)$ the class of all these Jensen measures.
Obviously, $\mu(D)=1$ for every $\mu\in{\sf J}_{z_{0}}(D)$. For $\mu\in{\sf
J}_{z_{0}}(D)$, the function

$V_{\mu}\colon z\longmapsto\int\ln|z^{\prime}-z|\operatorname{\,{\rm
d}}\mu-\ln|z|,\quad z\in\mathbb{C}\setminus z_{0},\quad V(\infty):=0,$ (8)

is the logarithmic potential of $\mu\in{\sf J}_{z_{0}}(D)$ with pole $z_{0}$
for $D$.

A positive subharmonic function $V\in\operatorname{{\sf
sbh}}^{+}(\mathbb{C}_{\infty}\setminus z_{0})$ is a Jensen potential for $D$
with pole $z_{0}\in D$ if there is a subset $S_{V}\Subset D$ such that
$V(z)=0$ for each $z\in D\setminus S_{V}$ and

$\limsup_{z_{0}\neq z\to z_{0}}\frac{V(z)}{-\ln|z-z_{0}|}\leq 1.$

We denote by ${\sf PJ}_{z_{0}}(D)$ the class of all these Jensen potentials.
If $D^{\prime}\Subset D$ be a subdomain in domain $D\subset\mathbb{C}$, then
its Green’s function ${\sf g}_{D^{\prime}}(\cdot,z_{0})$ with pole $z_{0}\in
D^{\prime}$ belongs to ${\sf PJ}_{z_{0}}(D)$.

######  Proposition 1 ((Kha03, , Propositions 1.2, 1.4)).

Let $D\neq\varnothing$ be a domain in $\mathbb{C}$ and $z_{0}\in D$.

  1. (i)

The mapping $\mathcal{P}\colon\mu\overset{\eqref{Vmeas}}{\longmapsto}V_{\mu}$
is an affine bijection from ${\sf J}_{z_{0}}(D)$ on ${\sf PJ}_{z_{0}}(D)$, and

$\mathcal{P}^{-1}(V)=\varDelta_{V}\bigm{|}_{\mathbb{C}\setminus
z_{0}}+\biggl{(}1-\limsup_{z_{0}\neq z\to
z_{0}}\frac{V(z)}{-\ln|z-z_{0}|}\biggr{)}\delta_{z_{0}},\quad V\in{\sf
PJ}_{z_{0}}(D).$ (9)

where $\delta_{z_{0}}$ is the Dirac probability measure with the support
$\operatorname{supp}\delta_{z_{0}}=z_{0}$.

  2. (ii)

If $\mu\in{\sf J}_{z_{0}}(D)$, then the following Poisson – Jensen formula
holds:

$u(z_{0})=\int_{D\setminus z_{0}}u\operatorname{\,{\rm
d}}\mu-\int_{D}V_{\mu}\operatorname{\,{\rm d}}\varDelta_{u}\quad\text{for each
$u\in\operatorname{{\sf sbh}}(D)$ with $u(z_{0})\neq-\infty$.}$ (10)

We denote by ${\sf PJ}_{z_{0}}^{\uparrow}(D):=\bigl{(}{\sf
PJ}_{z_{0}}(D)\bigr{)}^{\uparrow}$ the class of all test Jensen functions.

If $D\subset\mathbb{C}$ is a domain with non-polar boundary $\partial D$, then
there is its Green’s function ${\sf g}_{D}(\cdot,z_{0})$ with pole $z_{0}\in
D$, and ${\sf g}_{D}(\cdot,z_{0})$ is the largest Jensen test function in
${\sf PJ}_{z_{0}}^{\uparrow}(D)$. If the boundary $\partial D$ is polar, then
the largest test Jensen function in ${\sf PJ}_{z_{0}}^{\uparrow}(D)$ is the
$(+\infty)$-function $\bm{+\infty}\colon z\underset{z\in
D}{\longmapsto}+\infty$.

###  2.3 Main result on necessary conditions for zero subsets in domains

The main task of this section is to establish the largest possible range of
necessary conditions for the distribution of zero subsets of holomorphic
functions $f\in\operatorname{{\sf Hol}}(D)$ satisfying the upper constraint
$\ln|f|\leq M$ on $D$. We establish these conditions for arbitrary domains in
$\mathbb{C}$ and arbitrary $d$-subharmonic majorants $M$ from (6).

######  Theorem 1 (necessary conditions).

Let ${\sf Z}$ be a locally finite distribution of points in a domain
$D\subset\mathbb{C}$ and let $M$ be a function (6). Suppose that there exists
a function $f\in\operatorname{{\sf Hol}}(D)$ such that $f\neq 0$, $f({\sf
Z})=0$ and

$\ln\bigl{|}f(z)\bigr{|}\leq M(z)\quad\text{at each $z\in D$}.$ (11)

Then, for any closed set $S\Subset D$ with $\operatorname{{\sf
int}}S\neq\varnothing$ and for any $b\in\mathbb{R}^{+}$, there is a number
$C\in\mathbb{R}^{+}$ such that, for each test upper positive function
$v\overset{\eqref{v}}{\in}\operatorname{{\sf sbh}}_{0}^{+\uparrow}(D\setminus
S;\leq b)$, we have

$\sum_{{\sf z}_{j}\in D\setminus S}v({\sf z}_{j})\leq\int_{D\setminus
S}v\operatorname{\,{\rm d}}\varDelta_{M}+C\quad\text{provided $v$ is
$\varDelta_{M}$-summable on $D\setminus S$.}$ (12)

If $z_{0}\in D$, $f(z_{0})\neq 0$ and $M_{\operatorname{\text{\rm\tiny
up}}}(z_{0})+M_{\operatorname{\text{\rm\tiny low}}}(z_{0})\neq-\infty$, then
there is $C\in\mathbb{R}$ such that (12) holds with the singleton
$S:=\\{z_{0}\\}$ for each test Jensen function $v\in{\sf
PJ}_{z_{0}}^{\uparrow}(D)$.

###### Proof.

In the case $v\overset{\eqref{v}}{\in}\operatorname{{\sf
sbh}}_{0}^{+}(D\setminus S;\leq b)$ without $\uparrow$ we use

######  Lemma 1 ((KhaRoz18, , Main Theorem)).

For any point $z_{0}\in\operatorname{{\sf int}}S$ satisfying
$M_{\operatorname{\text{\rm\tiny up}}}(z_{0})+M_{\operatorname{\text{\rm\tiny
low}}}(z_{0})\neq-\infty$, any regular (for the Dirichlet problem) domain
$\tilde{D}\subset\mathbb{C}$ with the Green function ${\sf
g}_{\tilde{D}}(\cdot,z_{0})$ with a pole at $z_{0}\in\tilde{D}$ which
satisfies the conditions $S\Subset\tilde{D}\subset D$ and
$\mathbb{C}_{\infty}\setminus\operatorname{{\sf
clos}}\tilde{D}\neq\varnothing$, any function $u\in\operatorname{{\sf
sbh}}_{*}(D)$ satisfying the inequality $u\leq M$ on $D$, and any test
function $v\in\operatorname{{\sf sbh}}_{0}^{+}(D\setminus S;\leq b)$ the
following inequality holds:

$Cu(z_{0})+\int_{D\setminus S}v\operatorname{\,{\rm
d}}\varDelta_{u}\leq\int_{D\setminus S}v\operatorname{\,{\rm
d}}\varDelta_{M}+\int_{\tilde{D}\setminus S}v\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}}^{-}+C\overline{C}_{M},$
(13)

where $C:=b/\inf\limits_{z\in\partial S}{\sf g}_{\tilde{D}}(z,z_{0})>0$, and
the value $+\infty$ is possible for

$\overline{C}_{M}:=\int_{\tilde{D}\setminus z_{0}}{\sf
g}_{\tilde{D}}(\cdot,z_{0})\operatorname{\,{\rm
d}}\varDelta_{M}+\int_{\tilde{D}\setminus S}{\sf
g}_{\tilde{D}}(\cdot,z_{0})\operatorname{\,{\rm
d}}\varDelta_{M}^{-}+M^{+}(z_{0}),$ (14)

but for $\tilde{D}\Subset D$, this is a certain constant
$\overline{C}_{M}<+\infty$ independent of $v$ and $u$.

We put $u\overset{\eqref{q}}{:=}\ln|f|$ and choose a point
$z_{0}\in\operatorname{{\sf int}}S$ and a regular domain $\tilde{D}\Subset D$
with $f(z_{0})\neq 0$ and the required properties in Lemma 1. Then, in view of
(13)–(14), $C\in\mathbb{R}$, $\overline{C}_{M}\in\mathbb{R}$, and

$\int_{\tilde{D}\setminus S}v\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}}^{-}\leq
b\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}^{-}\bigl{(}\tilde{D}\bigr{)}<+\infty$

are independent of $f$ and $v$. Thus, there is a number $\tilde{C}$ such that

$\sum_{{\sf z}_{j}\in D\setminus S}v({\sf z}_{j})=\int_{D\setminus
S}v\operatorname{\,{\rm d}}n_{\sf Z}\leq\int_{D\setminus
S}v\operatorname{\,{\rm d}}n_{\operatorname{{\sf Zero}}_{f}}=\int_{D\setminus
S}v\operatorname{\,{\rm d}}\varDelta_{\ln|f|}\\\ =\int_{D\setminus
S}v\operatorname{\,{\rm d}}\varDelta_{u}\leq\int_{D\setminus
S}v\operatorname{\,{\rm
d}}\varDelta_{M}+\tilde{C}-C\ln\bigl{|}f(z_{0})\bigr{|}$

for each test function $v\in\operatorname{{\sf sbh}}_{0}^{+}(D\setminus S;\leq
b)$. Hence, for
$C^{\prime}:=\tilde{C}-C\ln\bigl{|}f(z_{0})\bigr{|}\in\mathbb{R}$, we obtain

$\sum_{{\sf z}_{j}\in D\setminus S}v({\sf z}_{j})+\int_{D\setminus
S}v\operatorname{\,{\rm d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}\leq\int_{D\setminus S}v\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny up}}}}+C^{\prime}$ (15)

for each test function $v\in\operatorname{{\sf sbh}}_{0}^{+}(D\setminus S;\leq
b)$. Let $(v_{k})_{k\in\mathbb{N}}\subset\operatorname{{\sf
sbh}}_{0}^{+}(D\setminus S;\leq b)$ be a increasing sequence and
$v:=\lim\limits_{k\to\infty}v_{k}\in\operatorname{{\sf
sbh}}_{0}^{+\uparrow}(D\setminus S;\leq b)$ is $\varDelta_{M}$-summable. Then

$\int_{D\setminus S}v_{k}\operatorname{\,{\rm d}}(n_{\sf
Z}+\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}})=\sum_{{\sf z}_{j}\in
D\setminus S}v_{k}({\sf z}_{j})+\int_{D\setminus S}v_{k}\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}}\\\
\overset{\eqref{Nf}}{\leq}\int_{D\setminus S}v_{k}\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}+C^{\prime}\leq\int_{D\setminus S}v\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}+C^{\prime}<+\infty\quad\text{for each $k\in\mathbb{N}$}.$

Applying the monotone convergence theorem for integrals to the left-hand side,
we obtain

$\sum_{{\sf z}_{j}\in D\setminus S}v({\sf z}_{j})+\int_{D\setminus
S}v\operatorname{\,{\rm d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}=\int_{D\setminus S}v\operatorname{\,{\rm d}}(n_{\sf
Z}+\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}})\leq\int_{D\setminus
S}v\operatorname{\,{\rm d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}+C^{\prime}$

for each test upper function $v\in\operatorname{{\sf
sbh}}_{0}^{+\uparrow}(D\setminus S;\leq b)$.

It remains to consider the case of test Jensen functions.

By the Weierstrass Theorem there are a function $f_{\sf
Z}\in\operatorname{{\sf Hol}}(D)$ with zero set $\operatorname{{\sf
Zero}}_{f_{\sf Z}}={\sf Z}$ and a function $g\in\operatorname{{\sf Hol}}(D)$
such that $f_{\sf Z}(z_{0})\neq 0$, $g(z_{0})\neq 0$ and $\ln|f_{\sf
Z}|+M_{\operatorname{\text{\rm\tiny
low}}}\overset{\eqref{q}}{\leq}M_{\operatorname{\text{\rm\tiny up}}}-\ln|g|$
on $D$ outside some polar set, and hence everywhere. Integrating with respect
to a Jensen measure $\mu\overset{\eqref{Jm}}{\in}J_{z_{0}}(D)$, we obtain

$\int_{D}\ln|f_{\sf Z}|\operatorname{\,{\rm
d}}\mu+\int_{D}M_{\operatorname{\text{\rm\tiny low}}}\operatorname{\,{\rm
d}}\mu\leq\int_{D}M_{\operatorname{\text{\rm\tiny up}}}\operatorname{\,{\rm
d}}\mu-\int_{D}\ln|g|\operatorname{\,{\rm
d}}\mu\leq\int_{D}M_{\operatorname{\text{\rm\tiny up}}}\operatorname{\,{\rm
d}}\mu-\ln\bigl{|}g(0)\bigr{|}.$

By the Poisson – Jensen formula of Proposition 1(ii), (10), to $\ln|f|$,
$M_{\operatorname{\text{\rm\tiny up}}}$, $M_{\operatorname{\text{\rm\tiny
low}}}$, we have

$\int_{D}V_{\mu}\operatorname{\,{\rm d}}n_{\sf
Z}+\ln\bigl{|}f(z_{0})\bigr{|}+\int_{D}V_{\mu}\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}+M_{\operatorname{\text{\rm\tiny
low}}}(z_{0})\leq\int_{D}V_{\mu}\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}+M_{\operatorname{\text{\rm\tiny up}}}(z_{0})-\ln\bigl{|}g(0)\bigr{|}$

Hence

$\int_{D}V_{\mu}\operatorname{\,{\rm d}}n_{\sf
Z}+\int_{D}V_{\mu}\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}\leq\int_{D}V_{\mu}\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}+\underset{C}{\underbrace{\Bigl{(}M_{\operatorname{\text{\rm\tiny
up}}}(z_{0})-\ln\bigl{|}f(z_{0})\bigr{|}-M_{\operatorname{\text{\rm\tiny
low}}}(z_{0})-\ln\bigl{|}g(0)\bigr{|}\Bigl{)}}}$

for logarithmic potentials $V_{\mu}$ of all Jensen measures $\mu\in{\sf
J}_{z_{0}}(D)$. By Proposition 1(i), if $\mu$ runs through ${\sf
J}_{z_{0}}(D)$, then $V_{m}$ runs through the class ${\sf PJ}_{z_{0}}$. Thus,

$\int_{D}V\operatorname{\,{\rm d}}\bigl{(}n_{\sf
Z}+\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}\bigr{)}=\int_{D}V\operatorname{\,{\rm d}}n_{\sf
Z}+\int_{D}V\operatorname{\,{\rm d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}\leq\int_{D}V\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny up}}}}+C\quad\text{for each
$V\in{\sf PJ}_{z_{0}}(D)$}.$

Let $(V_{k})_{k\in\mathbb{N}}\subset{\sf PJ}_{z_{0}}(D)$ be increasing and
$V:=\lim\limits_{k\to\infty}V_{k}\in{\sf PJ}_{z_{0}}^{\uparrow}(D)$ be
$\varDelta_{M}$-summable. Then

$\int_{D}V_{k}\operatorname{\,{\rm d}}\bigl{(}n_{\sf
Z}+\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}\bigr{)}\leq\int_{D}V_{k}\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}+C\leq\int_{D}V\operatorname{\,{\rm
d}}\varDelta_{M_{\operatorname{\text{\rm\tiny up}}}}+C<+\infty.$

Applying the monotone convergence theorem for integrals to the left-hand side,
we obtain

$\sum_{{\sf z}_{j}\in D\setminus S}V({\sf z}_{j})+\int_{D\setminus
S}V\operatorname{\,{\rm d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
low}}}}=\int_{D\setminus S}V\operatorname{\,{\rm d}}(n_{\sf
Z}+\varDelta_{M_{\operatorname{\text{\rm\tiny low}}}})\leq\int_{D\setminus
S}V\operatorname{\,{\rm d}}\varDelta_{M_{\operatorname{\text{\rm\tiny
up}}}}+C^{\prime}$

for each test Jensen function $V\in{\sf PJ}_{z_{0}}^{\uparrow}(D)$. ∎

##  3 Sufficient conditions for zero subsets of holomorphic functions with
upper constraints in domains

###  3.1 Integral means of subharmonic and $d$-subharmonic functions

We denote by
$D(z,t):=\bigl{\\{}z^{\prime}\in\mathbb{C}\colon|z^{\prime}-z|<t\bigr{\\}}$,
$\overline{D}(z,t):=\bigl{\\{}z^{\prime}\in\mathbb{C}\colon|z^{\prime}-z|\leq
t\bigr{\\}}$,
$\partial\overline{D}(z,t):=\overline{D}(z,t)\\!\setminus\\!D(z,t)$ an open
disk, a closed disk, a circle of radius $t\in\overline{\mathbb{R}}^{+}$
centered at $z\in\mathbb{C}$, respectively. If $D\neq\varnothing$ be a proper
domain in $\mathbb{C}$, i.e. $D\neq\mathbb{C}$, then we use a function
$r\colon D\to\mathbb{R}$ on $D$ such that

$\displaystyle\begin{cases}0\leq r(z)<\operatorname{{\sf dist}}(z,\partial
D):=\inf\limits_{z^{\prime}\in\partial D}|z-z^{\prime}|\quad\text{for each
$z\in D$},\\\ \inf\limits_{z\in K}r(z)>0\quad\text{for each $K\Subset
D$},\end{cases}$ (16D) but if $D=\mathbb{C}$, then we use another function
$\displaystyle
r(z)\underset{z\in\mathbb{C}}{:=}\frac{1}{(1+|z|)^{P}}\quad\text{with a number
$P\in\mathbb{R}^{+}$}.$ (16$\mathbb{C}$)

Let $u\colon D\to\overline{\mathbb{R}}$ be a function. The integral means of
$u$ over circles $\partial\overline{D}\bigl{(}z,r(z)\bigr{)}$ are denoted by

$\displaystyle u^{\odot r}(z)$
$\displaystyle:=\frac{1}{2\pi}\int_{0}^{2\pi}u\bigl{(}z+r(z)e^{i\theta}\bigr{)}\operatorname{\,{\rm
d}}\theta,\quad z\in D,\quad\overline{D}(z,r(z))\subset D,$ (17$\odot$) the
integral means of $u$ over disks $D\bigl{(}z,r(z)\bigr{)}$ are denoted by
$\displaystyle u^{\bullet r}(z)$ $\displaystyle:=\frac{1}{\pi
r^{2}(z)}\int_{0}^{r(z)}\int_{0}^{2\pi}u(z+te^{i\theta})\operatorname{\,{\rm
d}}\theta\,t\\!\operatorname{\,{\rm d}}t,\quad z\in D,\quad D(z,r(z))\subset
D,$ (17$\bullet$)

under the assumption that the integrals are well defined here, and (Rans, ,
2.6), Beardon , (FM, , Theorem 3)

$u\leq u^{\bullet r}\leq u^{\odot r}\leq u^{\bullet(\sqrt{e}r)}\quad\text{on
$D$\/ for each $u\in\operatorname{{\sf sbh}}_{*}(D)$},$ (18)

where the last inequality is given under the assumption that
$\sqrt{e}r<\operatorname{{\sf dist}}(\cdot,\partial D)$ on $D$.

We impose one very weak requirement on function (16D). For the function

$\displaystyle\widehat{r}(z)$
$\displaystyle:=\inf\biggl{\\{}R\in\mathbb{R}^{+}\Bigm{|}\bigcup_{z^{\prime}\in
D(z,r(z))}D\bigl{(}z^{\prime},r(z^{\prime})\bigr{)}\subset
D(z,R)\biggr{\\}},\quad z\in D,$ (19r) we require
$\displaystyle\overline{D}\bigl{(}z,\widehat{r}(z)\bigr{)}\subset
D\quad\text{for each $z\in D$.}$ (19$\rm\widehat{r}$)

We define the class (KhaKha19, , (1.12))

$\operatorname{{\sf sbh}}_{00}^{+}(D\setminus S;\leq
b):=\Bigl{\\{}v\in\operatorname{{\sf sbh}}_{0}^{+}(D\setminus S;\leq
b)\Bigm{|}\\\ \text{there is a subset $S_{v}\Subset D$ such that $v(z)=0$ at
each $z\in D\setminus S_{v}$}\Bigr{\\}}.$ (20)

of test subharmonic positive compactly supported functions for $D$ outside of
$S\Subset D$.

###  3.2 Main result on sufficient conditions for zero subsets in domains

The order of formulating sufficient conditions in Theorem 2 differs from the
order of formulating necessary conditions in Theorem 1. First, we formulate
sufficient conditions for arbitrary domains $D$ and $d$-subharmonic majorants
$M$ in terms of smooth Jensen potentials from ${\sf PJ}_{z_{0}}(D)$, and then
we formulate sufficient conditions for arbitrary domains $D$ with non-polar
boundary $\partial D$ and arbitrary $d$-subharmonic majorants $M$ from (6) in
terms of smooth test subharmonic functions from $\operatorname{{\sf
sbh}}_{0}(D\setminus S;\leq 1)$. The main task of this section is to establish
the smallest possible set of sufficient conditions for the distribution of
zero subsets of holomorphic functions $f\in\operatorname{{\sf Hol}}(D)$
satisfying the upper constraint $\ln|f|\leq M$ on $D$.

######  Theorem 2 (sufficient conditions).

Let ${\sf Z}$ be a locally finite distribution of points in an domain
$D\subset\mathbb{C}$ containing $z_{0}\notin{\sf Z}$ and $M$ be a
$d$-subharmonic function (6) with $M_{\operatorname{\text{\rm\tiny
up}}}(z_{0})+M_{\operatorname{\text{\rm\tiny low}}}(z_{0})\neq-\infty$.

If, there is a subdomain $U_{z_{0}}\Subset D$ containing $z_{0}\in U_{z_{0}}$
such that the inequality (12) with $S:=\\{z_{0}\\}$ is fulfilled for each
smooth Jensen potential

$\displaystyle v\in{\sf PJ}_{z_{0}}(D)\bigcap\operatorname{{\sf
har}}(U_{z_{0}}\setminus z_{0})\bigcap C^{\infty}(D\setminus z_{0})$ (21P)
satisfying $\displaystyle v(z)=-\ln|z-z_{0}|+O(1)\quad\text{as $z_{0}\neq z\to
z_{0}$},$ (210)

then, for each function (16) with (19) and with a number $P\in\mathbb{R}^{+}$
if $D:=\mathbb{C}$ and for any number $a>0$, there exists a function
$f\in\operatorname{{\sf Hol}}(D)$ such that $f\neq 0$, $f({\sf Z})=0$ and

$\displaystyle\ln|f|$
$\displaystyle\overset{\eqref{{M*r0}o}}{\leq}M_{\operatorname{\text{\rm\tiny
up}}}^{\odot\widehat{r}}-M_{\text{\tiny\rm low}}+R\quad\text{on $D$, where
$\widehat{r}$ is defined in \eqref{cup} and}$ (22M) $\displaystyle R(z)$
$\displaystyle:=\begin{cases}\ln\dfrac{1}{r(z)}+(1+a)\ln\bigl{(}1+|z|\bigr{)}\quad\text{if
$D\neq\mathbb{C}$},\\\ \ln\dfrac{1}{r(z)}\quad\text{if $D\neq\mathbb{C}$ is
simply connected or\/ $\complement\operatorname{{\sf
clos}}D\neq\varnothing$},\\\ 0\quad\text{if
$D=\mathbb{C}$},\end{cases}\quad\text{at each $z\in D$.}$ (22R)

In addition, if the boundary $\partial D$ of $D$ is non-polar and there exist
a closed subset $S\Subset D$ with $\operatorname{{\sf int}}S\neq\varnothing$
and a number $C\in\mathbb{R}^{+}$ such that the inequality (12) is fulfilled
for each smooth test function $v\overset{\eqref{v00}}{\in}\operatorname{{\sf
sbh}}_{00}^{+}(D\setminus S;\leq 1)\bigcap C^{\infty}(D\setminus S)$, then,
for any function (16D), there is a function $f\in\operatorname{{\sf Hol}}(D)$
such that $f\neq 0$, $f({\sf Z})=0$ and (22) is fulfilled.

###### Proof of Theorem 2.

We first prove the statement for Jensen potentials. We denote by ${\sf
PJ}_{z_{0}}^{1}(D)$ the class of all Jensen potentials $v$ satisfying (210)
and put (cf. (MenKha20, , (13V)))

$\mathcal{V}(U_{z_{0}}):={\sf PJ}_{z_{0}}^{1}(D)\bigcap\operatorname{{\sf
har}}(U_{z_{0}}\setminus z_{0})\bigcap C^{\infty}(D\setminus z_{0}).$ (23)

We denote by $\operatorname{{\sf Meas}}^{+}_{\infty}(D)$ the subclass of all
measures $\mu\in\operatorname{{\sf Meas}}^{+}(D)$ with a density $m\in
C^{\infty}(D)$, i.e. $\operatorname{\,{\rm d}}\mu=m\operatorname{\,{\rm
d}}\lambda$, where $\lambda$ is the Lebesgue measure on $D$, and put (cf.
(MenKha20, , (13M)))

$\mathcal{M}(U_{z_{0}}):={\sf J}_{z_{0}}(D)\bigcap\operatorname{{\sf
Meas}}_{\operatorname{cmp}}(D\setminus U_{z_{0}})\bigcap\operatorname{{\sf
Meas}}^{+}_{\infty}(D).$ (24)

By Proposition 1(i), it is easy to see that the of mapping $\mathcal{P}^{-1}$
in (9) defines a bijection of subclass $\mathcal{V}(U_{z_{0}})$ on subclass
$\mathcal{M}(U_{z_{0}})$ (MenKha20, , Theorem A).

By the Weierstrass Theorem there is a function $f_{\sf Z}\in\operatorname{{\sf
Hol}}(D)$ with zero set $\operatorname{{\sf Zero}}_{f_{\sf Z}}={\sf Z}$.

We have the inequality (12) with $S:=\\{z_{0}\\}$ for all Jensen potentials
$v\in\mathcal{V}(U_{z_{0}})$. Hence, by the Poisson – Jensen formula of
Proposition 1(ii), (10), applied to $\ln|f|$, $M_{\operatorname{\text{\rm\tiny
up}}}$, $M_{\operatorname{\text{\rm\tiny low}}}$, and by bijection
${\mathcal{P}}^{-1}(D)\colon\mathcal{V}(U_{z_{0}})\overset{\eqref{rto}}{\longrightarrow}\mathcal{M}(U_{z_{0}})$,
we have (cf. (MenKha20, , (15)))

$\int_{D}\underset{u}{\underbrace{\bigl{(}\ln|f_{\sf
Z}|+M_{\operatorname{\text{\rm\tiny low}}}\bigr{)}}}\operatorname{\,{\rm
d}}\mu=\int_{D}\ln|f_{\sf Z}|\operatorname{\,{\rm
d}}\mu+\int_{D}M_{\operatorname{\text{\rm\tiny low}}}\operatorname{\,{\rm
d}}\mu\\\ \leq\int_{D}M_{\operatorname{\text{\rm\tiny
up}}}\operatorname{\,{\rm
d}}\mu+\underset{c}{\underbrace{\Bigl{(}\ln\bigl{|}f_{\sf
Z}(z_{0})\bigr{|}+M_{\operatorname{\text{\rm\tiny
low}}}(z_{0})-M_{\operatorname{\text{\rm\tiny
up}}}(z_{0})\Bigr{)}}}\quad\text{for each $\mu\in\mathcal{M}(U_{z_{0}})$}.$
(25)

######  Lemma 2 (A very special case of (KhaRozKha19, , Corollary 8.1.II.1)
with $H:=\operatorname{{\sf sbh}}_{*}(D)$, cf. (MenKha20, , Theorem B)).

If for some number $c\in\mathbb{R}$ assertion (25) holds, then, for any
function $r$ satisfying (16D), there are a function $h\in\operatorname{{\sf
sbh}}_{*}(D)$ and a positive function $\check{r}\leq r$ from the class
$C^{\infty}(D)$ such that

$u+h\leq M_{\operatorname{\text{\rm\tiny up}}}^{\circledast\check{r}}\in
C^{\infty}(D)\quad\text{on $D$},$ (26)

where by construction (KhaRozKha19, , (8.3–6), (8.10)), (KhaKha19, ,
(2.18–19)) $M_{\operatorname{\text{\rm\tiny up}}}^{\circledast\check{r}}$ are
‘‘moving contracting’’ smoothing averages over some probabilistic measures
$\alpha^{(\check{r}(z))}\in\operatorname{{\sf
Meas}}^{+}_{\infty}\bigl{(}\overline{D}(z,\check{r}(z))\bigr{)}$, obtained by
the shift, compression, and normalization od a single approximate unit $a\in
C^{\infty}(\mathbb{C})$, depending on the modulus $|\cdot|$ only with support
$\operatorname{supp}a\subset\overline{D}(0,1)$.

By Lemma 2 we choose a subharmonic function $h\in\operatorname{{\sf
sbh}}_{*}(D)$ such that

$\ln|f_{\sf Z}|+M_{\operatorname{\text{\rm\tiny low}}}+h\leq
M_{\operatorname{\text{\rm\tiny up}}}^{\circledast\check{r}}\quad\text{on
$D$.}$

By (BaiTalKha16, , Proposition 3) or (KhaLob, , Theorem 4), we have
$M_{\operatorname{\text{\rm\tiny up}}}^{\circledast\check{r}}\leq
M_{\operatorname{\text{\rm\tiny up}}}^{\odot\check{r}}\leq
M_{\operatorname{\text{\rm\tiny up}}}^{\odot r}$ on $D$ for subharmonic
function $M_{\operatorname{\text{\rm\tiny up}}}$. Thus

$\ln|f_{\sf Z}|+M_{\operatorname{\text{\rm\tiny low}}}+h\leq
M_{\operatorname{\text{\rm\tiny up}}}^{\odot r}\quad\text{on $D$.}$

Hence

$\ln|f_{\sf Z}|+M_{\operatorname{\text{\rm\tiny low}}}+h^{\bullet
r}\overset{\eqref{{M*r0}b}}{\leq}\bigl{(}\ln|f_{\sf Z}|\bigr{)}^{\bullet
r}+M_{\operatorname{\text{\rm\tiny low}}}^{\bullet r}+h^{\bullet
r}\leq\bigl{(}M_{\operatorname{\text{\rm\tiny up}}}^{\odot r}\bigr{)}^{\bullet
r}\quad\text{on $D$.}$ (27)

######  Lemma 3 ((KhaBai16, , Theorem 3, Corollary 3(i),(iii))).

Let $h\in\operatorname{{\sf sbh}}_{*}(D)$ be a subharmonic function on a
domain $D\subset\mathbb{C}$. Then, for any number $a>0$, there is a function
$g\in\operatorname{{\sf Hol}}(D)$ such that $g\neq 0$ and

$\ln|g|\leq h^{\bullet t}+R\quad\text{on $D$,}$ (28)

where $R$ is a function from (22R).

By Lemma 3 and (27), we get

$\ln|f_{\sf Z}g|+M_{\operatorname{\text{\rm\tiny low}}}=\ln|f_{\sf
Z}|+M_{\operatorname{\text{\rm\tiny
low}}}+\ln|g|\leq\bigl{(}M_{\operatorname{\text{\rm\tiny up}}}^{\odot
r}\bigr{)}^{\bullet r}+R\quad\text{on $D$,}$ (29)

where $f:=f_{\sf Z}g\neq 0$ and $f({\sf Z})=0$.

The following lemma is an elementary very special case of (KhaLob, , Theorems
2, 4).

######  Lemma 4.

If $r$ and $\widehat{r}$ are functions (19), then $(u^{\odot r})^{\bullet
r}\leq u^{\odot\widehat{r}}$ on $D$.

By Lemma 4, it follows from (29) that

$\ln|f|+M_{\operatorname{\text{\rm\tiny low}}}=\ln|f_{\sf
Z}g|+M_{\operatorname{\text{\rm\tiny low}}}\leq
M_{\operatorname{\text{\rm\tiny up}}}^{\odot\widehat{r}}+R\quad\text{on $D$.}$

Thus, we have proved (22) under (12) for the smooth Jensen potentials.

Let us now consider the case of a domain $D$ with non-polar boundary $\partial
D$.

######  Lemma 5 ((KhaKha19, , Theorem 3)).

Under the conditions of Theorem 2, there is a subharmonic function
$u\in\operatorname{{\sf sbh}}_{*}(D)$ such that $n_{\sf Z}\leq\varDelta_{u}$
and $u\leq M^{\bullet r}\quad\text{on $D$}$.

By Lemma 5 there is a subharmonic function $u\in\operatorname{{\sf
sbh}}_{*}(D)$ such that

$n_{\sf Z}\leq\varDelta_{u}\quad\text{and}\quad u\leq M^{\bullet
r}=M_{\operatorname{\text{\rm\tiny up}}}^{\bullet r}-M_{\text{\rm\tiny
low}}^{\bullet r}\overset{\eqref{Ml}}{\leq}M_{\operatorname{\text{\rm\tiny
up}}}^{\bullet r}-M_{\text{\rm\tiny low}}\quad\text{on $D$}.$ (30)

By the Weierstrass Theorem, there is $f_{\sf Z}\in\operatorname{{\sf Hol}}(D)$
with $\operatorname{{\sf Zero}}_{f_{\sf Z}}={\sf Z}$, and $f_{\sf Z}\neq 0$.

Consider a $d$-subharmonic function $h:=u-\ln|f_{\sf Z}|$ with Riesz charge
$\varDelta_{h}=\varDelta_{u}-n_{\sf Z}\overset{\eqref{in:hM}}{\geq}0$, i. e.,
$\varDelta_{h}\in\operatorname{{\sf Meas}}^{+}(D)$ and $h\in\operatorname{{\sf
sbh}}_{*}(D)$. It follows from (30) that

$\ln|f_{\sf
Z}|+h=u\overset{\eqref{in:hM}}{\leq}M_{\operatorname{\text{\rm\tiny
up}}}^{\bullet r}-M_{\text{\rm\tiny low}}\quad\text{on $D$}.$ (31)

Hence, for $\ln|f_{\sf Z}|\in\operatorname{{\sf sbh}}_{*}(D)$ and
$h\in\operatorname{{\sf sbh}}_{*}(D)$, we get

$\ln|f_{\sf Z}|+h^{\bullet r}\leq\bigl{(}\ln|f_{\sf Z}|\bigr{)}^{\bullet
r}+h^{\bullet r}=\bigl{(}\ln|f_{\sf Z}|+h\bigr{)}^{\bullet r}=u^{\bullet r}\\\
\overset{\eqref{m1}}{\leq}\bigl{(}M_{\operatorname{\text{\rm\tiny
up}}}^{\bullet r}\bigr{)}^{\bullet r}-M_{\text{\rm\tiny low}}^{\bullet
r}\overset{\eqref{Ml}}{\leq}\bigl{(}M_{\operatorname{\text{\rm\tiny
up}}}^{\bullet r}\bigr{)}^{\bullet r}-M_{\text{\rm\tiny low}}\quad\text{on
$D$}.$ (32)

Using Lemma 3 we set $f:=f_{\sf Z}g\neq 0$. Then $f({\sf Z})=0$ since ${\sf
Z}=\operatorname{{\sf Zero}}_{f_{\sf Z}}$, and

$\ln|f|=\ln|f_{\sf Z}g|=\ln|f_{\sf
Z}|+\ln|g|\overset{\eqref{ln}}{\leq}\ln|f_{\sf Z}|+h^{\bullet
r}+R\overset{\eqref{lnf}}{\leq}\bigl{(}M_{\operatorname{\text{\rm\tiny
up}}}^{\bullet r}\bigr{)}^{\bullet r}-M_{\text{\rm\tiny low}}+R\quad\text{on
$D$}.$ (33)

By Lemma 4 with $u:=M_{\operatorname{\text{\rm\tiny up}}}$ we obtain
$\ln|f|\overset{\eqref{lnf+}}{\leq}M_{\operatorname{\text{\rm\tiny
up}}}^{\odot\widehat{r}}-M_{\text{\rm\tiny low}}+R$ on $D$. ∎

##  4 Zero subsets in the complex plane

In this section, we give the results of Theorems 1 and 2 a form related to
subharmonic functions of polynomial growth and point out a very general case
when the necessary and sufficient conditions coincide. We denote by

${\rm\sf Pot}:=\Bigl{\\{}p\in\operatorname{{\sf
sbh}}_{*}(\mathbb{C})\Bigm{|}\limsup_{z\to\infty}\frac{p(z)}{\ln|z|}<+\infty\Bigr{\\}}$

the convex cone over $\mathbb{R}^{+}$ of all subharmonic functions of
polynomial growth (Hayman, , 6.7.2). We use the convex subcone over
$\mathbb{R}^{+}$

${\rm\sf Pot}_{0}^{+1}:=\left\\{p\in{\rm\sf Pot}\Bigm{|}p(0)=0,\quad p\geq
0\text{ on $\mathbb{C}$},\quad\limsup_{z\to\infty}\frac{p(z)}{\ln|z|}\leq
1\right\\}\subset{\rm\sf Pot}$ (34)

of positive subharmonic functions of polynomial growth with unit upper
seminormization at $\infty$.

######  Theorem 3.

Let ${\sf Z}$ be a locally finite distribution of points in $\mathbb{C}$ and
$0\notin{\sf Z}$. Let $M$ be a $d$-subharmonic function (6) on
$D:=\mathbb{C}$. Suppose that $M_{\text{\rm\tiny low}}(0)\neq-\infty$ and
there are numbers $P\in\mathbb{R}^{+}$ and $C\in\mathbb{R}^{+}$ such that (3)
holds. Then the following three statements are equivalent:

  1. I.

There exists an entire function $f\neq 0$ such that $f({\sf Z})=0$ and

$\ln\bigl{|}f(z)\bigr{|}\leq M(z)\quad\text{at each $z\in\mathbb{C}$}.$ (35)

  2. II.

There is a number $C\in\mathbb{R}^{+}$ such that, for each $p\in{\rm\sf
Pot}_{0}^{+1}$, we have

$\sum_{j}p\Bigl{(}\frac{1}{\bar{\sf
z}_{j}}\Bigr{)}\leq\int_{\mathbb{C}}p\Bigl{(}\frac{1}{\bar{z}}\Bigr{)}\operatorname{\,{\rm
d}}\varDelta_{M}(z)+C\quad\text{provided $p\Bigl{(}\frac{1}{\bar{z}}\Bigr{)}$
is $\varDelta_{M}$-summable on $\mathbb{C}\setminus 0$.}$ (36)

  3. III.

There are numbers $C\in\mathbb{R}^{+}$ and $R_{0}>0$ such that (36) is
fulfilled for each

$\displaystyle p\in{\rm\sf Pot}_{0}^{+1}\bigcap
C^{\infty}(\mathbb{C})\bigcap\operatorname{{\sf
har}}\bigl{(}\mathbb{C}\setminus\overline{D}(0,R_{0})\bigr{)}$ (37P) such that
$p=0$ on some neighborhood of the origin and $\displaystyle
p(z)=\ln|z|+O(1)\quad\text{as $z\to\infty$}.$ (370)

###### Proof.

Here (35) is (11) for $D=\mathbb{C}$. Let $z_{0}:=0$. The inversion
transformation

$z\underset{z\in\mathbb{C}}{\longmapsto}\frac{1}{\bar{z}},\quad
0\mapsto\infty\mapsto 0$

from $\mathbb{C}_{\infty}$ onto $\mathbb{C}_{\infty}$ gives a bijection from
${\sf Pot}_{0}^{+1}$ onto ${\sf PJ}_{0}^{\uparrow}(\mathbb{C})$ and a
bijection from the class (37) onto the class (21), or onto the class
$\mathcal{V}(U_{z_{0}})$ from (23). Thus, (36) is (12) for
$v(z)\underset{z\in\mathbb{C}_{\infty}}{=}p(1/\bar{z})$.

Hence the implication I$\Longrightarrow$II follows from Theorem 1, the
implication II$\Longrightarrow$III is obvious, and the implication
III$\Longrightarrow$I follows from Theorem 2 if we take into account condition
(3). ∎

Acknowledgments. The work was supported by a Grant of the Russian Science
Foundation (Project No. 18-11-00002).

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