The representation of a meromorphic function as the quotient of entire functions and\\ Paley problem in ${\bC}^n$: survey of some results
B.~N. Khabibullin \thanks{Supported by the Russian Foundation of Basic Research under Grant No.~00-01-00770.}
{Dedicated to Professor N.~I.~Akhiezer in the year of his 100th anniversary}
The classical representation problem for a meromorphic function $f$ in $\bC^n$, $n\ge 1$, consists in representing $f$ as the quotient $f=g/h$ of two entire functions $g$ and $h$, each with logarithm of modulus majorized by a function as close as possible to the Nevanlinna characteristic. Here we introduce generalizations of the Nevanlinna characteristic and give a short survey of classical and recent results on the representation of a meromorphic function in terms such characteristics. When $f$ has a finite lower order, the Paley problem on best possible estimates of the growth of entire functions $g$ and $h$ in the representations $f=g/h$ will be considered. Also we point out to some unsolved problems in this area.
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\title{The representation of  a meromorphic function
as the quotient of entire functions and\\
Paley problem in ${\bC}^n$: survey of some
results}  %%\footnote{Unpublished}}%%32A15, 32A20, 31A05, 31C10
\author{B.~N. Khabibullin
\thanks{Supported  by the Russian Foundation of Basic Research
under Grant No.~00-01-00770.}\\
\small{Dedicated to Professor N.~I.~Akhiezer in the year
of his 100th anniversary}}
\date{}

\begin{document}
\maketitle
\begin{abstract}
The classical representation problem for a meromorphic function $f$
in  $\bC^n$, $n\ge 1$,  consists in representing $f$ as the quotient $f=g/h$
of two entire functions $g$ and $h$, each with logarithm of
modulus majorized by a function as close as possible to the
Nevanlinna characteristic. Here we introduce  generalizations of the
Nevanlinna characteristic and give a short survey of classical and recent
results on the representation of a meromorphic function  in terms such
characteristics. When $f$ has a finite lower order, the Paley problem
on  best possible estimates of the growth of entire functions
$g$ and $h$ in the representations $f=g/h$  will be considered.
Also we point out to  some unsolved problems in this area.
\end{abstract}


\pagestyle{myheadings}
\markboth{B.~N. KHABIBULLIN}{THE REPRESENTATION OF A MEROMORPHIC
FUNCTION...}


\subsection*{Introduction}

Let $f$ be a meromorphic function  (we write
$f\in \Mer$)  in the complex plane.
At the beginning of the last century the definition of the
classic Nevanlinna characteristic  $T(r; f)$ and the Liouville Theorem
on the growth of entire functions were giving
an one of first results on the representation of a meromorphic
function as the quotient of entire functions.
\begin{Theorem}[J.~Liouville--R.~Nevanlinna]\label{LN}
A meromorphic function $f$ in the complex plane can be represented
as the quotient $f=g/h$ of polinomials  $g$ and $h$, i.e.
$$
\log \bigl( |g(z)|+|h(z)|\bigr) \leq O( \log r)\, , \quad r\to +\infty
\, ,
$$
iff
$$
T(r; f)\le O( \log r)\,  , \quad r\to +\infty \, .
$$
\end{Theorem}

The classic Nevanlinna Theorem on the representation of a meromorphic
function $f$ of bounded characteristic  $T(r; f)\le \const \,$, $r<1$,
in unit disk $\D$ as the quotient of bounded holomorphic functions in $\D$
can be considered also as one of sources of our survey.

A first essential result for transcendental meromorphic functions
follows immediately from the Lindel\"of Theorem of the beginning
of XX century (see, for example, \cite{Rubel}, \cite{GO}):

\begin{Theorem}[E.~Lindel\"of]\label{Li}
A  meromorphic function $f$ in the complex plane can be represented
as the quotient $f=g/h$ of entire functions  $g$ and $h$
of finite type for order $\rho$, i.e.
$$
\log \bigl( |g(z)|+|h(z)|\bigr) \leq O( r^{\rho})\, , \quad r\to
+\infty \, ,
$$
iff
$$
T(r; f)\le O(  r^{\rho})\,  , \quad r\to +\infty \, .
$$
\end{Theorem}

Here we consider  far-reaching generalizations of
these classic results and  give a  survey of basic and
recent results on the {\it problem of the representation\/}
of a meromorphic function in $\bC^n$
as the quotient of entire functions in terms of  various characteristics.
Also we point out to  some unsolved problems in this area.

Let $f \in \Mer$ in $\bC^n$ and
\begin{equation}
\label{repr}
f=\frac{g_f}{h_f}
\end{equation}
be a {\it canonical  representation\/} of $f$ as the quotient of
entire functions $g_f$ and $h_f$ which are locally relatively prime, i.e.
at each  point $z\in \bC^n$, where $g_f(z)=h_f(z)=0$, the functions
$g$ and $h$ are relatively prime in the ring of germs of functions analytic at
$z$. In $\bC^n$ such representations always exist \cite{GunnR},
\cite{Hormander}. Both functions
\begin{subequations}
\label{uf}
\begin{align}
u_f &=\max \bigl\{ \log |g_f|, \log |h_f|\bigr\} \label{ufi}  \\
\notag \text{ or } \qquad &\\
u_f &=\log \bigl( |g_f|^2 + |h_f|^2 \bigr)^{1/2}  \label{ufii}
\end{align}
\end{subequations}
are plurisubharmonic on $\bC^n$.
Consideration of these {\it characteristic functions\/} $u_f$ is useful and
natural, since  $u_f$ is used to define the various types
of Nevanlinna characteristic in particular in the Shimizu-Ahlfors form etc.
\cite{GO}, \cite{Hayman},  \cite{Kondratyuk}, \cite{Kujala},
\cite{Khabib93-I}, \cite{Skoda}--\cite{Skoda72}, \cite{Stoll}--\cite{Taylor}.


A function $g$ on $\bC^n$ is said to be {\it circular\/}
if $g(e^{i\theta}z)=g(z)$ for all $z\in \bC^n$ and $\theta \in \R$.

First we introduce the {\it circular Nevanlinna characteristic}
\begin{equation}
\label{Tcirc}
{T_f^c}(z)=\frac1{2\pi} \int\limits_{0}^{2\pi} u_f(e^{i\theta}z)\, d\theta \, ,
\quad z\in \bC^n\, ,
\end{equation}
which is a circular plurisubharmonic function \cite{Kujala}.
For example, if  $f(0)=1$ and $T_f(t; \zeta )$, $t\geq 0$,
are the Nevanlinna characteristics
(in the Shimizu-Ahlfors form for the case \eqref{ufii}) \cite{GO}
of the family of the meromorphic functions $f_{\zeta}(w)=f(w\zeta )$
of one variable $w\in \bC$, where $\zeta$ runs through the unit sphere
$S^n \subset \bC^n$, then $T_f(t; \zeta )\leq {T_f^c}(z)$,
$z=te^{i\theta}\zeta$. Equality holds
if $n=1$ or if $g_f$ and $h_f$ do not have common zeros ($n>1$).

Let $\mathbf{\mbox{\L}} =(L_1, \ldots , L_k)$  be a
fixed simply ordered collection of complex vector subspaces
in ${\bC}^n$ and ${\bC}^n$ is the direct sum
of $L_k$. As usual, we write
\begin{equation}
\label{directsum}
\bC^n = L_1\oplus \cdots \oplus L_k\, .
\end{equation}

Denote by $\R_+$ the set of nonnegative numbers.

%%Throughout this review, for $z\in \bC^n$, we denote
Let $z\in \bC^n$  and
\begin{subequations}\label{z}
%%\begin{split}
%%\label{z}
\begin{align}
z&=w_1+\cdots +w_k, \qquad  w_p\in L_p,\quad p=1, \ldots , k, \label{zz}\\
r_p&=|w_p|,\quad \vec r\, =(r_1, \ldots , r_k)\in \R_+^k, \label{zw}
\end{align}
\end{subequations}
where $|w_p|$ is the Euclidean norm $w_p$ in $L_p$.
For every $z\in \bC^n$  the representation \eqref{zz} is unique.

If $u(z)$ is a function on $\bC^n$,
then we consider   the function $u$ also as the function
$u(z)=u(w_1, \ldots , w_k)$  on the direct sum \eqref{directsum}
under the  notations \eqref{z}. We keep also the
same notation $u$ for $u(w_1, \ldots , w_k)$.

We introduce {\it Nevanlinna\/} $\mathbf{\mbox{\L}}$-{\it characteristic\/} of
$f\in \Mer$ in the form
$$
T_{\mathbf{\mbox{\L}}}(\vec{r}\, ; f)=\int\limits_{S_1} \cdots \int\limits_{S_k}
u_f(r_1\zeta_1, \ldots , r_k\zeta_k)
\, ds^{(1)} (\zeta_1)\cdots ds^{(k)} (\zeta_k),
\quad \vec{r}\, \in \R_+^k \, ,
%%\end{equation}
$$
where the functions $u_f$ are defined in \eqref{uf},
and $ds^{(p)}$ is the area element on the unit sphere $S_p$ in $L_p$.

The {\it Nevanlinna\/} $\mathbf{\mbox{\L}}$-{\it characteristic\/} can
be defined also in the form
\begin{equation}
\label{T}
T_{\mathbf{\mbox{\L}}}(\vec{r}\, ; f)=\int\limits_{S_1} \cdots \int\limits_{S_k}
T_f^c(r_1\zeta_1, \ldots , r_k\zeta_k)
\, ds^{(1)} (\zeta_1)\cdots ds^{(k)} (\zeta_k).
\end{equation}
\newcommand{\Fl}[3]{#3_{#1},\ldots ,#3_{#2}}
\newcommand{\Fc}[3]{#3_{#1}\cdots #3_{#2}}

It is easy to check that every Nevanlinna
$\mathbf{\mbox{\L}}$-characteristic is defined by \eqref{repr}--\eqref{uf}
to within an additive  constant.

The Nevanlinna $\mathbf{\mbox{\L}}$-characteristic gives the various variants
of the classical Nevanlinna characteri\-s\-t\-ics of $f$ in $\bC^n$.
So, if $k=1$ and $L_1=\bC^n$ in \eqref{directsum},
then we get Nevanlinna characteristic $T(r; f)$, $r\geq 0$,
by exhaustion of $\bC^n$ by balls $r\B^n$, where $\B^n$ is the unit ball in
$\bC^n$; if $k=n$ and $L_1=\cdots =L_n=\bC$ in \eqref{directsum},
then we get Nevanlinna characteristic $T(\vec r\, ; f)$,
$\vec r\, \in \R_+^n$, by exhaustion of $\bC^n$ by  polydisks
$P^n(\vec r\, )=\{z\in \bC^n: |w_p|=r_p, p=1, \ldots , n \}$
with the preceding notations \eqref{z}
(see  \cite{Stoll}--\cite{Taylor}, \cite{Kujala}, \cite{Skoda}--\cite{Skoda72},
\cite{Khabib93-I} for various variants of definitions of such
Nevanlinna characteristics).

Below we formulate the well-known and recent main results on the
representation of a meromorphic function on $\bC^n$, $n\ge 1$.

The author expresses deep gratitude to
I.~F.~Krasichkov-Ternovski\u{\i} for useful discussion
of this review and a correction of the text, and to M.~L.~So\-din,
A.~A.~Kondratyuk and  Ya.~V.~Vasyl'kiv for a valuable information,
and also to  the reviewer for helpful remarks.

\section{The representation  of arbitrary\\ meromorphic functions}

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\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\addtocounter{footnote}{+1}

A continuous increasing nonnegative function $\lambda $ on $\R_+$
is called a {\it growth function\/} (see \cite{Taylor-Rubel},
\cite{Miles} and \cite{Kujala}).

For $n=1$ and  for a {\it meromorphic function $f$ of  finite $\lambda$-type},
i.e. satisfying the condition
\begin{equation}
\label{ballestm}
T(r; f)\le \const \cdot \lambda (r)+\const \, , \quad r>0\, ,
\end{equation}
where $\lambda $ has either  the {\it slow growth\/}, i.e. satisfies
the condition
\begin{equation}
\label{slow1}
\lambda (2r)\le O\bigl(\lambda (r)\bigr) \,  ,\quad r \to +\infty\; ,
\end{equation}
or  $\log \lambda (\exp t)$ is {\it convex\/}, L.~A.~Rubel and B.~A.~Taylor
\cite[Theorems 5.4, 3.5-6, 3.2]{Taylor-Rubel} indicated (1968)
{\it a construction of entire functions $g$ and $h$ such that $f=g/h$
and
\begin{equation}\label{balleste}
\log \bigl(|g(z)|+|h(z)|\bigr)\le A\lambda (Br)+C \; ,\quad r=|z|>0\; ,
\end{equation}
where $A$, $B$ and C are constants\/}.

All conditions for $\lambda$  was removed for $n=1$ (1970-72) by
J.~B.~Miles in \cite{MilesB}, \cite{Miles}
(see also the original proof of J.~B.~Miles
in \cite[Ch.~4]{Kondratyuk}, \cite[Ch.~14]{Rubel}):


\begin{Theorem}[L.~A.~Rubel--B.~A.~Taylor--J.~B.~Miles]
\label{RTM}
 Every fu\-n\-c\-t\-i\-on
$f\in \Mer$ in the complex plane $\bC$ can be represented as the quotient
 $f=g/h$ of entire functions $g$ and $h$ such that
\begin{equation}
\label{ballesteM}
\log \bigl(|g(z)|+|h(z)|\bigr)\le AT(Br; f)+C \; ,\quad r=|z|>0\; ,
\end{equation}
where $A$, $B$ and $C$ are
constants\footnote{The importance of such
results for $n>1$ was marked by W.~Stoll in his book  \cite[p.~85]{Stoll85}:
{\it `` Unfortunately the theorem of Miles  {\rm \cite{Miles}} has not been
proved for several variables''}.}.
\end{Theorem}

Analogous results for functions in the half-plane
and in the unit disk can be find  in \cite{Mlt} and in \cite{Beck} recp..

The following special case of the representation theorems
was established for $n=1$ in \cite{Gol'dberg} (1972):
\begin{Theorem}[A.~A. Gol'dberg]
\label{Gol}
Every meromorphic function $f$ in the complex plane
${\bC}$ can be re\-p\-r\-e\-s\-e\-n\-ted
as the quot\-i\-ent $f=g_1/g_2$ of entire functions $g_1$ and $g_2$
\underline{without  common zeros} such that
$$
\log T(r; g_k)=o\bigl(T(r; f)\bigr) \, , \quad r\to \infty \; ,\quad k=1,2\; .
$$
\end{Theorem}

This theorem (for functions $g_1$ and $g_2$ {\it without common zeros})
is, in a certain sense,  best possible (see also \cite{Gol'dberg}).
Further results in this direction (\underline{without  common zeros})
can to find  in \cite{Skoda3}, \cite{Berg}, \cite{VSh}, \cite{KonVas}.

Our last one-dimensional  result \cite{Khabib01} (2001) is the following
\begin{Theorem}[B.~N. Khabibullin]
\label{Khab01}
Let $f$ be meromorphic function in the complex plane.
For each  nonincreasing convex function $\e (r)>0$ in $\R_+$
entire functions $g$ and $h$ in the representation $f=g/h$
can be chosen so that the estimate
\begin{equation}
\label{Te}
\log \bigl(|g(z)|+|h(z)|\bigr)\le
\frac{A_{\e}}{\e (r)} T\bigl((1+\varepsilon (r) )\cdot r\, ; f\bigr)
+B_{\e}\, , \quad r=|z|>0 \, ,
\end{equation}
holds. Here $A_{\e}$ and $B_{\e}$ are constants depending on the function
$\e (r)$ (but independent of $r$).
\end{Theorem}

In particular for  \, $1+\e (r) \equiv B>1$ in \eqref{Te} the last
Theorem \ref{Khab01} implies the Rubel--Taylor--Miles Theorem \ref{RTM},
and for \, $\e (r) =\epsilon \bigm/(1+r) $ in \eqref{Te} with a constant
$\epsilon >0$ we obtain the sharpening
of the one-dimensional variant of Skoda Theorem  \ref{Sko}
with $(1+ r)$ in place of $(1+r)^3$ in \eqref{Skoest}.
Besides the Theorem \ref{Khab01} can be  essentially stronger
than the Rubel--Taylor--Miles  Theorem or the one-dimensional Skoda Theorem
if  the Nevanlinna characteristic $T(r; f)$ has
rapid growth and the decreasing function $\e (r)$  is chosen in conformity
with the growth of $T(r; f)$.

For $n>1$  the first result on the representation problem
for a special growth functions $\lambda (r)=r^{\rho }$
(that solves the problem of L.~A.~Rubel \cite[Problem 8]{Problems} (1968))
is apparently due to W.~Stoll  \cite[Proposition 6.1]{Stoll} (1968).

Let $\lambda (\vec{r}\, )=\lambda (\Fl{1}{n}{r})$,
$\, \vec{r}\,  \in \R_+^n$, be a positive
continuous function which is nondecreasing in each variable,
and $T(\vec{r}\, , f)\leq \lambda (\vec{r}\, )$, $\, \vec{r}\, \in
\R_+^n$. B.~A.~Taylor  posed the problem\footnote{In the original,
``{\it When is $\Lambda$ the field of quotients of
${\Lambda}_{E}\,$\/}?''} \cite[p.~470]{Taylor} (1968):
\begin{description}
\item[{\bf (TP)}]
{\it when can a meromorphic function $f$ be represented as the quotient
$f=g/h$ of entire functions $g$ and $h$ such that
\begin{equation}
\label{estim2}
\log \bigl(|g(z)|+|h(z)|\bigr)\le A\lambda (B\vec{r}\, )+C \, ,
\; r_j=|z_j|\ge 1 \, , \; 1\leq j \leq n\, ,
\end{equation}
where $A$, $B$ and $C$ are constants ?\/}
\end{description}

He proved the following Theorem  \cite[Theorem]{Taylor} (1968):
\begin{Theorem}[B.~A.~Taylor]
\label{Tay}
If $\lambda (\vec{r}\, )$ is
slowly increasing in each variable, in the sense that
\begin{equation*}
\lambda (r_1, \ldots , 2r_j, \ldots , r_n)\le
A_j\lambda (\Fl{1}{n}{r})
\end{equation*}
for some constant $A_j>0$, $1\le j \le n$, and
$T(\vec{r}\, ; f)\le \const \cdot \lambda (\vec{r}\, )$
for all $r_j \ge 1$, $1\le j \le n$, then
there are entire functions $g$ and $h$ such that  $f=g/h$ and
\eqref{estim2} holds for all $r_j \ge 1$.
\end{Theorem}

Further progress for meromorphic functions $f$ in several variables and
for the characteristic $T(r; f)$, i.e. $k=1$,
under  special conditions on the behavior of $\lambda$,
 was made in \cite{KujalaB}, \cite[Propositions 7.3, 9.10]{Kujala} (1969-71):
\begin{Theorem}[R.~O.~Kujala]
\label{Kuj}
If there are constants $a_0$ $($resp., a vanishing function $a_0=a_0(r)$,
$r\to +\infty$$)$,
$a$, $b$ and $R$ in $\R_+$  and $p_0$ in $\N$ such that
\begin{equation*}
\int\limits_s^r \lambda(t)t^{-p-1}\, dt\le a_0\lambda (br)r^{-p}
+a\lambda (bs)s^{-p}
\end{equation*}
whenever $r\ge s >R$ and $p_0\le p$ in $\N$  $($in particular,
if $\lambda$ satisfies the slow growth condition \eqref{slow1}\/$)$, then
under the condition  \eqref{ballestm}
the meromorphic function  $f$ in $\bC^n$ can be represented as the quotient
$f=g/h$ of entire functions $g$ and $h$ such that
\eqref{balleste} $($recp., with a vanishing function $A=A(r)$, $r\to
+\infty$$)$ is valid.
\end{Theorem}

Except the Theorems \ref{LN}, \ref{Li}, \ref{Gol}, \ref{Khab01} and the
mentioned result for $\lambda (r)=r^{\rho}$ from W.~Stoll \cite{Stoll},
all these results were obtained by the Fourier series method
(see  \cite{Kondratyuk}, \cite{Kujala}, \cite{Miles},
\cite{Noverraz}, \cite{Taylor-Rubel}--\cite{Rubel}, \cite{Taylor}).
This method was used first by N.~I.~Akhiezer in \cite{Akhiezer} (1927)
for the new proof  the Lindel\"of Theorem and
later  by L.~A.~Rubel in \cite{Rubel1} (1961) (see in this
connection also \cite[P.~85--88]{GO}).

Using the $\bar \partial$-problem method, H.~Skoda
\cite{Skoda}--\cite{Skoda72} (1971-72) obtained  the following
results (see also \cite[Theorem 9.12]{Stoll85}):
\begin{Theorem}[H.~Skoda]
\label{Sko}
For every  constant $\epsilon >0$  entire functions $g$ and $h$
in representation $f=g/h\in \Mer$ can be chosen to
satisfy the estimates
\begin{equation}
\label{Skoest}
\log \bigl(|g(z)|+|h(z)|\bigr)
\le C(\epsilon ,s)(1+r)^{4n-1}T(r+\epsilon ; f)
\end{equation}
or
\begin{equation}
\label{Skoest1}
\log \bigl(|g(z)|+|h(z)|\bigr)\le C(\epsilon , s)
\bigl(\log (1+r^2)\bigr)^2 T\bigl((1+\epsilon )r; f\bigr)
\end{equation}
for all $r=|z|\geq s>0$,
where $C(\epsilon , s)$ is a constant depending on $\epsilon $ and
$s$.
\end{Theorem}

It should be noted  \cite[p.~295]{Stoll85} that the pair $(g, h)$
in the Theorem \ref{Sko} can have a common divisor, can depend on
$\varepsilon$ but can not depend on $s$, and can be different in
\eqref{Skoest1} from the pair chosen in \eqref{Skoest}. The case
\eqref{Skoest} is good for rapid growth, the case \eqref{Skoest1}
is good for slow growth.

An analysis  of these  results indicates a definite
rift between the cases of slow and rapid growth of the majorants
$\lambda$ or of the characteristics $T(\cdot ; f)$, both in methods
(the Fourier series method in the Theorems \ref{RTM}, \ref{Tay}--\ref{Kuj}
and the $\bar \partial$-problem method in the Theorem \ref{Sko})
and  the estimates and conditions on $\lambda$ or $T(\cdot
; f)$. Our balayage (or sweeping out) method (1990-93) enables
 us to get rid of this rift and yields results in a complete form.

To be more specific, first this balayage method was applied
for a new short nonconstructive proof of Rubel-Taylor-Miles Theorem
\ref{RTM}  in \cite[\S~3]{Khabib91} (1991).
The improvement of this method for $n=1$ made it possible
recently to prove the Theorem \ref{Khab01}.
In our articles \cite{Khabib92}, \cite{Khabib93-I}
we established  for functions of several variables
 the following results that are, in a certain sense, extreme  relative  to
the Nevanlinna characteristics $T(r; f)$, $r\geq 0$, and \eqref{Tcirc}.

\begin{Theorem}[B.~N.~Khabibullin]
\label{repres}
Let $f\in \Mer$, $n\ge 1$. Then
\begin{description}
\item[(T)]
{\rm {\cite[Theorem 1.3]{Khabib92}}, {\cite[Theorem 5]{Khabib93-I}}
(1992-93)}

For each constant $\e >0$  the function $f$ can be represented
as the quotient
\begin{equation} \label{gh}
f=\frac{g}{h}\, , \qquad g \text{ and } h \text{ are entire functions},
\end{equation}
such that
\begin{equation}
\label{estT}
\log \bigl(|g(z)|+|h(z)|\bigr)\le A_{\e}T\bigl((1+\e )r; f\bigr)+C_{\e}\, ,
\quad     r=|z|>0;
\end{equation}

\item[(Tc)]
{\rm {\cite[Theorem 4]{Khabib93-I}} (1993)}

Under the conditions  $T_f^c(z)\le \lambda (z)$,
$z\in \bC^n$, $f(0)=1$, where the function  $\lambda$  is circular
continuous positive increasing on all rays with origin at $0$
and satisfies the following two H\"ormander conditions\/
{\rm \cite{Hormander67}:}

\begin{description}
\item[(L)] \label{L}
$ \log \bigl(1+|z|\bigr)\leq O\bigl(\lambda (z)\bigr)\; \text{ as } \;
      |z|\to +\infty \,$,
and
\item[(H)] \label{Hor}  for any number
  $\epsilon >0$ there exist positive constans $c_1, c_2, c_3, c_4 $ such that
$$
|z-\zeta |\le \exp \bigl(-c_1\lambda (z)-c_2\bigr) \Longrightarrow \,
\lambda (\zeta)\le c_3\lambda\bigl((1+\epsilon )z\bigr)+c_4,
\; z\in \bC^n,
$$
or
\item[(\^H)]
for any $\epsilon  >0$ there exist positive constants $\sigma$, $c_1$, and
$c_2$ such that
\begin{equation}   \label{H2}
|z-\zeta |\le \sigma \, \Longrightarrow \,
\lambda (\zeta)\le c_1\lambda\bigl((1+\epsilon )z\bigr)+c_2,  \quad z\in \bC^n,
\end{equation}
\end{description}
for any $\e >0$  the function $f$ can be represented
as a quotient  \eqref{gh} such that
\begin{equation} \label{cTlambda}
\log \bigl(|g(z)|+|h(z)|\bigr)\le A_{\e} \lambda \bigl((1+\e ) z\bigr)
+ C_{\e}\; ,
\quad z\in {\bC}^n \; ,
\end{equation}
where under the condition \eqref{H2}
functions $g$ and $h$ can be chosen so that $g(0)=h(0)=1$.
\end{description}

Here $A_{\e}$ and $C_{\e}$ are constants depending on $\e$.
\end{Theorem}

Besides the fact that the extra conditions on the growth function
$\lambda$ was removed in  the Theorem \ref{repres} (Part {\bf (T)}),
 it also refines
the Theorems \ref{RTM}, \ref{Kuj} and \ref{Sko}. For example, the constant
$B$ in  \eqref{balleste} and \eqref{ballesteM}
is replaced by a constant $1+\e >1$
arbitrary close to $1$, and in \eqref{Skoest} and in \eqref{Skoest1}
it is possible to
remove the power factor before  $T(\cdot ; f)$
if in \eqref{Skoest} $r+\e$  is replaced by $(1+\e )r$, where $\e >0$.
Following Gol'dberg \cite{Gol'dberg}, we can show that we cannot set
$\e =0$ in general in \eqref{estT} and in \eqref{cTlambda}.
An analog of the Theorem \ref{repres} (Part {\bf (T)})
for $\delta$-subharmonic functions was established
by O.~V.~Veselovskaya \cite{Ves} (1984) by the Fourier series method.

If we have  some additional information on the
functions $g_f$ and $h_f$ in  the initial representation
\eqref{repr}, then these additional properties can be preserved
sometimes in the final representation \eqref{gh}
(see \cite[Theorems 1--3]{Khabib98} (1998)):

\begin{Theorem}[B.~N.~Khabibullin] If
it is known in addition to conditions of  Theorem \ref{repres} that for
$f\in \Mer$ with $f(0)=1$  there exists
the representation  \eqref{repr}, where $g_f$ and
$h_f$   are  \underline{bounded in some open set} $B\subset {\bC}^n$,
then in  {\rm {\bf (T)}}  $($recp., in {\rm {\bf (Tc)}}$)$
functions $g$ and $h$  can be chosen  so that
besides  \eqref{estT} $($recp., \eqref {cTlambda}$)$
for any $\gamma \geq 1$ the inequlity
$$
\log \bigl(|g(z)|+|h(z)|\bigr)\le C_{\gamma }\log \bigl(2+|z|\bigr)\; ,\quad
z\in B_{\gamma }\, ,
$$
holds, where
$$
B_{\gamma }=\{ z\in B : \; \dist (z,\partial B)\ge
\bigl(1+|z|\bigr)^{-\gamma}\}
\, ,
$$
$\dist (z,\partial B)$ is the distance from $z$ to the boundary
$\partial B$ of $B$,  and $C_{\gamma }$ is constant.

If $n=1$, then functions $g$ and $h$ in \eqref{gh} can be chosen
also so that $g$ and $h$ are bounded on the set
$\{z\in B : \dist (z,\partial B)\ge 1/{\gamma}\}$.
\end{Theorem}

For the general case of the Nevanlinna $\mathbf{\mbox{\L}}$-characteristic
we have at present  the following result \cite{Khabib00} (2000):
\begin{Theorem}[B.~N.~Khabibullin] \label{TL}
Let $f\in \Mer$ and $\bC^n$
is represented as \eqref{directsum}. Then,
with the preceding notations \eqref{uf}--\eqref{T},
for any constant $\e >0$,   there exists a representation
\eqref{gh} such that
\begin{equation} \label{TLest}
\log \bigl(|g(z)|+|h(z)|\bigr)
\le A_{\e} \text{\rm $T_{\mathbf{\mbox{\L}}}^+$}\bigl((1+\e )\vec r\, +
B_{\e}\cdot \vec 1 \, ; f\bigr)
+C_{\e}\log \bigl(2+| z|\bigr)
\end{equation}
for all $z\in \bC^n$, where  $A_{\e}, B_{\e}, C_{\e}$ are constants and
$\vec 1\, =(1, \ldots , 1) \in\R_+^k$, $\; T^+=\max \{ T, 0\}$.
\end{Theorem}

It is easy to see, that the Theorem \ref{TL} implies the Theorem
\ref{repres} (Part {\bf (T)}),
because always $\log \bigl(2+| z|\bigr)\leq O\bigl(T(r, f)\bigr)$,
$r=|z|$, $f\not\equiv 0, \infty$.
Also, if
\begin{equation} \label{log}
\log \bigl(2+|\vec r\,|\bigr)\leq O\bigl(\lambda (\vec{r}\,
)\bigr)\, , \quad  k=n, \quad |\vec r \, |=\sqrt{r_1^2+\cdots +r_k^2}\, ,
\end{equation}
then  the Theorem \ref{TL}  is the extension of Theorem \ref{Tay}.
It  solves the Taylor problem {\bf (TP)} with  minimal
estimate \eqref{log} from below for $\lambda (\vec{r}\, )$.

A general scheme of the solution of the representation
problem and some other problems (balayage method) was presented in
\cite{Khabib96}--\cite{Khabib97}, \cite{Khabib01I}.

\subsubsection*{Unsolved problems.}

\paragraph*{Problem 1.} Is it true,
that {\it the summand $C\log \bigl(2+| z|\bigr)$
in  \eqref{TLest} in the Theorem \ref{TL}
can be removed\/} ?

In any case it is true for $k=1$, and in the Theorem \ref{Tay} for $k=n$
if $\lambda (\vec r \, )$ is slowly increasing.

\paragraph*{Problem 2.}
{\it Extend the result of the Theorem \ref{Khab01}
to $\bC^n$, $n>1$, for classic Nevanlinna characteristic $T(r; f)$
and for Nevanlinna $\mbox{\L}$-characteristic.}

\paragraph*{Problem 3.}
Denote by   $\mathcal{PSH}_p$  the cone of all
{\it plurisubharmonic\/} function on $L_p$, $p=1, \ldots , k$,
where $L_p$ was defined in \eqref{directsum}.
Denote by  $\mathcal{M}_p^+$ the cone of all {\it positive Borel measures
$($Radon measures$)$ with compact support\/} in $L_p\, .$

We introduce in $\mathcal{M}_p^+$ the {\it partial order\/}
${\prec}$ by (see \cite[Ch.~XI]{Meyer})
$$
 (\; \nu \,{\prec} \,\mu\;  )\; \Longleftrightarrow \;
\Bigl(\;  \int v\, d\nu \leq \int v\, d\mu
\;  \text{ for all } \; v\in \mathcal{PSH}_p \; \Bigr)\, .
$$
If $\nu \prec \mu$, then the measure $\mu\in \mathcal{M}_p^+$,
is called a {\it balayage\/}
( also  {\it sweeping out\/} or {\it Jensen measure\/} \cite{Gamelin})
of the measure $\nu$ (with respect to the cone $\mathcal{PSH}_p\,$).

Further,  let
$\boldsymbol \mu =(\Fl{1}{k}{\mu})$
be a fixed simply ordered collection of measures, where  every measure
$\mu_p\in \mathcal{M}_p^+$ is  the balayage
of the Dirac measure $\delta_p \in \mathcal{M}_p^+$
at the point $0\in L_p$, i.e. $\int u \, d\delta_p =u(0)$.

Let $u_f$ be a plurisubharmonic function from \eqref{uf}.
A function
\begin{equation*}
T_{\mathbf{\mbox{\L}}, \boldsymbol \mu }(\vec r\, ; f)=
\int\limits_{S_1} \cdots \int\limits_{S_k} u_f(r_1\zeta_1, \ldots , r_k\zeta_k)
\, d\mu_1 ({\zeta}_1)\cdots d\mu_k ({\zeta}_k)\, ,
\quad  \vec r\, \in \R_+^k,
\end{equation*}
will be called the {\it Nevanlinna-Jensen\/}
$(\mathbf{\mbox{\L}}, \boldsymbol \mu )$-{\it characteristic\/} of  $f\in \Mer$.
{\it How can the Theorem \ref{TL} be extended  for the
Nevanlinna-Jensen
{\rm $(\mathbf{\mbox{\L}}, \boldsymbol \mu )$}-cha\-r\-a\-c\-t\-e\-r\-i\-s\-tic\/} ?



\section{The representation of meromorphic\\
functions with restrictions on the type\\
and on the circled indicator}

\setcounter{equation}{0}

In this section we consider the problem of the representation of a
meromorphic function $f$ in $\bC^n$ as a quotient \eqref{gh}
with  best possible estimates of the circled indicators and of the types
of the entire functions $g$ and $f$.

{\it The representation} (or {\it factorization}) {\it type
${\sigma}_n^*(f, \rho )$ of a meromorphic function $f$ on $\bC^n$ of  order
$\rho$} is defined to be the infimum of the numbers $\sigma$
for which there are entire functions $g$ and $h$ of
order $\rho$ with the type less
than $\sigma$  (see \cite{LG}) such that $f=g/h$.

Let
\begin{equation} \label{Prho}
P_1 (\rho )=\begin{cases}
\dfrac{\pi \rho}{\sin \pi \rho} \, ,
& \text{ if \; $0\leq \rho \leq 1/2$}\, , \\
\pi \rho \, , & \text{ if \; $\rho \, >\, 1/2$}\, .
\end{cases}
\end{equation}

For $n=1$ the first sharp result in this direction was established
apparently in our paper \cite[Theorem 2]{Khabib92t} (1992):

\begin{Theorem}[B.~N.~Khabibullin]  \label{Tn1}
Let $f$ be a meromorphic
function  in the complex plane $\bC$. If the type of the Nevanlinna
characteristic $T(r; f)$ is equal to $\sigma$ with order $\rho$, i.e.
\begin{equation} \label{Ttype}
\limsup_{r\to +\infty} r^{-\rho}T(r;f)=\sigma\, ,
\end{equation}
then the sharp estimate
$\;
\sigma \leq  {\sigma}_1^*(f, \rho )  \leq \sigma P_1 (\rho )
\;$
is valid.
\end{Theorem}
%%Let $u$ be a plurisubharmonic function on $\bC^n$, $n\geq 1$, of finite type
%%with order $\rho$.

For $n>1$ we set
\begin{equation} \label{Pnrho}
P_n (\rho )=P_1 (\rho )\, \prod_{k=1}^{n-1} \Bigl(1+\frac{\rho }{2k}\Bigr)\, .
\end{equation}

For $n>1$ we have \cite[Theorem 3]{Khabib93-II} (1993):
\begin{Theorem}[B.~N.~Khabibullin] \label{Tn}
If the type of the Nevanlinna
characteristic $T(r; f)$ is equal to $\sigma$ with order $\rho$, i.e.
 \eqref{Ttype} holds,
then
$$
\sigma \leq  {\sigma}_n^*(f, \rho )  \leq
\sigma P_1 (\rho ) \, \prod_{k=1}^{n-1}b\Bigl(\frac{\rho}{2k}\Bigr)
\leq e^{n-1}P_n(\rho ) ,
$$
where
\begin{equation}   \label{Pb}
b(x)=\begin{cases}
ex \, ,                  & \text{ if \; $x\geq 1$}\, , \\
e^x \,  ,                & \text{ if \; $x< 1$}\, ,
\end{cases}
\end{equation}
and for $\rho \leq 1$ the sharp estimate

\begin{equation}   \label{Psh}
\sigma \leq  {\sigma}_n^*(f, \rho )  \leq \sigma P_n (\rho )
\end{equation}
is valid.
\end{Theorem}

Let $u$ be a plurisubharmonic function on $\bC^n$ of finite type with
order $\rho$. Define the function
$$
h_{c\rho}(z, u)=\limsup_{\xi \to \infty} |\xi |^{-\rho}u(\xi z),
\quad \xi \in \bC \, , \; z\in \bC^n\, .
$$
Its upper semicontinuous regularization
$$
h_{c\rho}^*(z, u)=\limsup_{\zeta \to z} h_{c\rho}(\zeta , u)
$$
is called \cite[Definition 1.29]{LG} the {\it circled indicator} of
growth of $u$ (with order $\rho$).
If we are dealing with circled indicator of growth  $h_{c\rho}^* (\zeta , u)$,
then we assume a priori that $u$ is of finite type with order $\rho$.
It is known that the circled indicator is complex-homogeneous of
degree $\rho$, i.e., $h_{c\rho}^*(\xi z, u)=|\xi |^{\rho}
h_{c\rho}^*(z, u)$, $\xi \in \bC$, is plurisubharmonic, and
$h_{c\rho}^*(z, u)\geq 0$ if $u\not\equiv -\infty$.
We remind that the circular Nevanlinna characteristic
$T_f^c(z)$ of $f\in \Mer$ in \eqref{Tcirc} is plurisubharmonic.
Therefore, circled
indicator of growth for $T_f^c$ is defined.
In terms of the circled indicator of growth $h_{c\rho}^* (\zeta , T_f^c)$
we have the following best possible result \cite[Theorem 5.1, Example
5.1]{Khabib93-II} (1993):

\begin{Theorem}[B.~N.~Khabibullin]  \label{Thcirc}
Suppose that $f\in \Mer$, $f(0)=1$,
and $k$ is a continuous circular function on $S^n \subset \bC^n$.
\begin{enumerate}
\item If
$\quad
h_{c\rho}^* (\zeta , T_f^c) <k(\zeta ), \quad \zeta \in S^n\, , \,
$
then there exist entire functions $g$ and $h$ such that $f=g/h$,
$g(0)=h(0)=1$, and
$$
\max\bigl\{ h_{c\rho}^* (\zeta , g), h_{c\rho}^* (\zeta , h)\bigr\}
<P_1(\rho )k(\zeta),
\quad \zeta \in S^n\, .
$$
The constant $P_1(\rho )$ cannot in general be diminished, not
even for any one of the functions $g$ and $h$.
\item If $f=g/h$ is a representation of $f$ as the quotient of entire
functions  $g$ and $h$, then
$$
\max\bigl\{ h_{c\rho}^* (\zeta , g), h_{c\rho}^* (\zeta , h)\bigr\}\geq
h_{c\rho}^* (\zeta , T_f^c)\, , \quad \zeta \in S^n\, ,
$$
and this estimate is sharp.
\end{enumerate}
\end{Theorem}

The main difficulty consists in  the obtaining  the upper estimates
in the Theorems    \ref{Tn1}--\ref{Thcirc}. The success here
was achieved with the use of the balayage method (see the preceding
section).

\subsubsection*{Unsolved problems.}
\paragraph*{Problem 1.}
{\it  Prove  the upper estimate in \eqref{Psh} for all $\rho$\/}.
See the commentary to the last Problem  in the next section.

\paragraph*{Problem 2.} {\it Extend the results of the Theorems
\ref{Tn}--\ref{Thcirc} to
the Nevanlinna {\rm $\mathbf{\mbox{\L}}$}-characteristic
and to the the Nevanlinna-Jensen {\rm $(\mathbf{\mbox{\L}},
\boldsymbol \mu )$}-characteristic\/}.

\section{Paley problem}

\setcounter{equation}{0}

The subject of this section is close to the themes of the previous
sections in both  estimates and methods of proofs.

For a   function $u$ in $\bC^n$ (recp., in $\R^m$)
with the range of values $[-\infty , +\infty]$
or $\R \cup \{ \infty\}$, we set
\begin{equation*}
M(r; u)=\max \bigl\{ u(z): |z|=r\} \, , \quad
u^+(z)=\max \bigl\{ u(z), 0 \bigr\}\, .
\end{equation*}

In 1932 R.~Paley supposed that for any entire function $f$ of order
$\rho \geq 0$ the inequality
\begin{equation} \label{Paleycon}
\liminf\limits_{r\to +\infty}
\frac{M\bigl(r; \log |f|\bigr)}{T(r; f)}
\leq P_1(\rho )
\end{equation}
holds (see the definition of  $P_1(\rho )$ in \eqref{Prho}).

In \eqref{Paleycon} the equality is achieved, in particular, for the
Mittag-Leffler's function (see, for example,  \cite[p.111]{GO}).
Earlier the inequality \eqref{Paleycon} was proved by G.~Valiron
\cite{Valiron} (1930) and by A.~Whalund \cite{Whalund} (1929)
for $\rho \leq 1/2$.

The Paley conjecture was conclusively proved by N.~V.~Govorov
\cite{Govorov} only in 1967. In 1968, V.~P.~Petrenko \cite{Petrenko} extended
this result to meromorphic functions of finite lower order.
Different proofs  was obtained later by I.~V.~Ostrovski\u{\i}
\cite{Ostrovsk} for meromorphic
functions and by M.~R.~Ess\'en \cite{Essen} for subharmonic functions
in the complex plane.
\begin{Theorem}[N.~V.~Govorov--V.~P.~Petrenko] \label{GP}
If $f$ is a meromorphic function in the complex plane $\bC$
of finite lower order $\lambda$, i.e.,
\begin{equation} \label{lowerorder}
\lambda =\liminf_{r\to +\infty}
\frac{\log T(r; f)}{\log |r|} \, < \, +\infty \, ,
\end{equation}
then the inequality
\begin{equation} \label{Paleylambda}
\liminf\limits_{r\to +\infty}
\frac{M\bigl(r; \log |f|\bigr)}{T(r; f)}
\leq P_1(\lambda )
\end{equation}
holds.
\end{Theorem}

The relative growth of $T(r; f)$ and $M\bigl(r; \log |f|\bigr)$
for entire and meromorphic functions of infinite order
in the complex plane has also been  considered by many authors.
%%We found a detailed description  of investigation of this
%%case before 1994  in  \cite{BB}.

For entire functions of infinite order in the complex plane
in \cite{Chuang}(1981), \cite{MSh}(1984) and finally in
\cite{DDL}(1990--1991) was proved the following result.

\begin{Theorem}\label{DDL}
{\bf (C.~T.~Chuang, I.~I.~Marchenko--A.~I.~Shcherba,\linebreak
C.~J.~Dai--D.~Drasin--B.~Q.~Li)}
Let  $f$ be an entire function of infinite order in the complex plane.
If $\psi (x)$ is  increasing and positive for $x\geq x_0>0$
and if
\begin{equation}\label{psi}
\int_{x_0}^{\infty}\frac{dx}{\psi (x)}<\infty \, ,
\end{equation}
then
\begin{equation}\label{Tpsi}
\liminf_{r\to +\infty}\frac{M\bigl(r; \log |f|\bigr)}
{T(r; f) \cdot \psi\bigl( \log T(r; f)\bigr)}=0
\end{equation}
and even
\begin{equation*}\label{oTpsi}
M\bigl(r; \log |f|\bigr)=
o\Bigr( T(r; f) \cdot \psi\bigl( \log T(r; f)\bigr)\Bigr)
\end{equation*}
as $r\to +\infty$  through a set of logarithmic density one.
\end{Theorem}
In \cite{MSh} and \cite{DDL} it is also shown
that the results are best possible in some sense.

For meromorphic  functions
in \cite{DDL} (1990--1991) the following was obtained.
\begin{Theorem}[C.~J.~Dai-D.~Drasin-B.~Q.~Li]\label{DDLm}
Let $f$ be a meromorphic fu\-n\-c\-tion
of infinite order in the complex plane.
If $\psi (x)$ is the same as in the Theorem \ref{DDL}
then
\begin{equation*}\label{oTmpsi}
M\bigl(r; \log |f|\bigr)=
o\Bigr( T(r; f) \cdot \psi\bigl( \log T(r; f)\bigr)
\cdot \log \psi\bigl( \log T(r; f)\bigr)\Bigr)
\end{equation*}
as $r\to +\infty$  through a set of logarithmic density one.
\end{Theorem}

A different approach has been taken in \cite{Bergweiler}, \cite{BB}
(1990--1994) where the characteristic $M\bigl(r; \log |f|\bigr)$  has been
compared  with the derivative of $T(r; f)$.

\begin{Theorem}[W.~Bergweiler-H.~Block]\label{BBm}
Let $f$ be a meromorphic fun\-c\-tion of infinite order in the complex
plane. Then
\begin{equation*}\label{oTgam}
\liminf_{r\to +\infty}
\frac{M\bigl(r; \log |f|\bigr)}{r T_{-}'(r; f)}\leq \pi \, ,
\end{equation*}
where $T_{-}'(r; \cdot )$ is the left-side derivative of $T$ of $r$.

Let $\psi (x)$ be positive and continuously
diffirentiable for $x\geq x_0 >0$ such that $\psi (x)/x$
is non-decreasing, $\psi (x)\leq \sqrt{\psi (x)}$, and
\eqref{psi} is satisfied. Then \eqref{Tpsi} holds.
\end{Theorem}

The articles of I.~I.~Marchenko \cite{March1}-\cite{March2} contains
much  information in particular on the growth of entire and
meromorphic functions of infinite order.

We don't know any results for entire and meromorphic functions
of infinite order in $\bC^n$,
$n>1$, wich are analogs of the Theorems \ref{DDL}--\ref{BBm}.

Let $u$ be a subharmonic function in $\R^m$, $m\geq 2$,
(resp., in $\bC^n$)
and let $|\mathcal S^{m-1}|$ be the area of the unit sphere
$\mathcal S^{m-1}$ in $\R^m$, $d{s}$
is the area element on the unit sphere $\mathcal S^{m-1}$.
\begin{equation*}
m_q (r; u)=\biggl(\;  \frac1{|\mathcal S^{m-1}|}\,
\int\limits_{|\mathcal S^{m-1}|} \bigl|u(rx)\bigr|^q \,
d{s}(x) \,\biggr)^{1/q}\, ,
\quad r>0, \; 1\leq q <+\infty.
\end{equation*}
Then Nevanlinna characteristic $T(r;u)$ and $M(r; u)$ are respectively
$$
 T(r; u)=m_1 (r; u^+), \quad   M(r; u^+)=m_{\infty} (r; u^+)=
\lim_{q\to +\infty}m_q (r; u^+) .
$$
In particular if $f$ is a entire function in $\bC^n$ then
$T(r; f)=T\bigl(r; \log |f|\bigr)$, $m=2n$.

An analogue of \eqref{Paleylambda} for subharmonic functions
of finite lower order $\lambda$ in $\R^m$, $m\geq 3$, was
obtained by B.~Dahlberg \cite[Theorem 1.2]{Dahlberg}.

To be more specific suppose $\lambda \in (0, +\infty )$ is given.
The Gegenbauer functions $C_{\lambda}^{\gamma}$ are given as the solutions
of the differential equation
$$
(1-x^2)\, \frac{d^2u}{dx^2}-(2\gamma +1)x\, \frac{du}{dx}+\lambda
(\lambda+2\gamma )\, u=0, \quad -1<x<1\, ,
$$
with the normalization
$$
\lim_{x\to 1-0}C_{\lambda}^{\gamma} (x)=C_{\lambda}^{\gamma} (1)=
\frac{\Gamma (\lambda+2\gamma )}{\Gamma (2\gamma )\Gamma (\lambda +1)}.
$$
Put
$\;
a_{\lambda}=\sup \bigl\{ t: C_{\lambda}^{\frac{m-2}{2}} (t)=0\bigr\}
\; $
and define the function $u_{\lambda}$ in $\R^m$, $m\geq 3$, by
\begin{equation*}
u_{\lambda}(x)=\begin{cases}
0 \quad &\text{if $x_1\leq a_{\lambda}r$} \\
r^{\lambda}C_{\lambda}^{\frac{m-2}{2}} (x_1/r)
&\text{if $x_1> a_{\lambda}r$}\, ,
\end{cases}
\end{equation*}
where $x=(x_1, \ldots , x_m)$ and $r=|x|$.

The function $u_{\lambda}$ is subharmonic in $\R^m$ and the lower
order of $u_{\lambda}$ is $\lambda$. Set
\begin{equation*}
c(\lambda , m)=
\liminf_{r\to +\infty} \,
\frac{M(r; u_{\lambda})}{T(r; u_{\lambda })}\, .
\end{equation*}
\begin{Theorem}[B.~Dahlberg] \label{Dahl}
Let $u$ be a subharmonic function in
$\R^m$, $m\geq 3$, of finite lower order $\lambda >0$.
Then we have that
\begin{equation*}
\liminf_{r\to +\infty} \,
\frac{M(r; u )}{T(r; u)}\, \leq \, c(\lambda ,
m)\, ,
\end{equation*}
and this estimate is  best possible.
\end{Theorem}

The generalization of the Theorems \ref{GP}, \ref{Dahl}
is also known.

\begin{Theorem}[M.~L.~Sodin {\rm \cite{Sodin} (1983)}] \label{ThSodin}
Let $u$ be a subharmonic function
of lower order $\lambda$ in the complex plane. Then
\begin{equation} \label{mqSodin}
\liminf_{r\to +\infty}\frac{m_q (r; u^+)}{T(r; u)}\leq
m_q(S_{\lambda}), \quad 1 < q\leq +\infty,
\end{equation}
where $m_q(S_{\lambda})$ is the Lebesque mean of order $q$,
$1<q<+\infty$, of the function
\begin{equation*}
S_{\lambda}(\varphi )=\begin{cases} \pi \lambda \cos \lambda \varphi
\quad &\text{if $\quad |\varphi|\leq  \dfrac{\pi}{2\lambda}$},\\
0 \quad &\text{if $\quad \dfrac{\pi}{2\lambda}<|\varphi|\leq \pi $},
\end{cases}
\end{equation*}
for $\lambda \geq 1/2$ and
$$
S_{\lambda}(\varphi )= \frac{\pi \lambda \cos \lambda \varphi}{\sin
\pi \lambda}\, , \qquad |\varphi |\leq \pi,
$$
for $\lambda <1/2$, $ \; m_{\infty}(S_{\lambda})=
\max \bigl\{ S_{\lambda}(\varphi ): |\varphi |\leq \pi \bigr\}$.
The inequality \eqref{mqSodin} is sharp.
\end{Theorem}

This result was extended to $m\geq 3$ in \cite[Theorem 1]{KonTarVas}
(1995).

Set
\begin{equation*}
Q_{\lambda}^{\frac{m-2}{2}}(\theta )=\begin{cases}
A(\lambda, m)\, C_{\lambda}^{\frac{m-2}{2}}(\cos \theta )
\quad &\text{if $\quad 0\leq \theta \leq  \alpha_{\lambda}$},\\
0 \quad &\text{if $\quad \alpha_{\lambda}<\theta \leq \pi $},
\end{cases}
\end{equation*}
where
\begin{align*}
\alpha_{\lambda}&=\min \bigl\{\theta \in (0, \pi ):
C_{\lambda}^{\frac{m-2}{2}}(\cos \theta )=0 \bigr\}, \\
A(\lambda, m)&=\biggl( \; \frac{|\mathcal S^{m-2}|}{|\mathcal S^{m-1}|}
\, \int\limits_0^{\alpha_{\lambda}} \,
C_{\lambda}^{\frac{m-2}{2}}(\cos \theta ) {\sin}^{m-2}\theta \, d\theta
\biggr)^{-1}\, .
\end{align*}

\begin{Theorem}[A.~A.~Kondratyuk--S.~I.~Tarasyuk--Ya.~V.~Vasyl'kiv]
Let $u$ be a subharmonic function
of lower order $\lambda >0$ in $\R^m$, $m\geq 3$. Then
for every $q$, $1<q\leq +\infty$,  the inequality
\begin{equation} \label{mqKTV}
\liminf_{r\to +\infty}\frac{m_q (r; u^+)}{T(r; u)}\leq M_q({\lambda}, m)
\end{equation}
is true, where $M_q({\lambda}, m)=m_q
\bigl(Q_{\lambda}^{\frac{m-2}{2}}\, \bigr)$.
There exists a subharmonic function $u$ of lower order $\lambda >0$ in $\R^m$
for which in \eqref{mqKTV}  the equality is achieved.
\end{Theorem}

Besides, from W.~Hayman result \cite{Hayman} it follows that for
subharmonic functions of order $\lambda =0$ the inequlity
$$
\liminf_{r\to +\infty}\frac{m_q (r; u^+)}{T(r; u)}\leq 1, \quad
1<q\leq +\infty \, ,
$$
holds.

In \cite{Khabib95} (1995) we extended \eqref{Paleylambda} to
meromorphic functions $f$ in $\bC^n$, $n>1$. Instead of $M(r; \log |f|)$
the corresponding formulation involves in this case the family of
characteristics $M(r; \log |f_{\zeta}|)$, $\zeta \in S^n$, of the
slice functions $f_{\zeta}(w)=f(w\zeta )$, $w\in \bC$. This is
understanble, for in the case of a meromorphic function the quantity
$M(r; \log |f|)$ can be identically equal to $\infty$ starting from
some value of $r$, whereas the Nevanlinna characteristics of $f$ can
have very slow growth (one example is the function $f(z_1,
z_2)=(1+z_1)/(1+z_2)$ in $\bC^2$).
\begin{Theorem}[B.~N.~Khabibullin] \label{Khabmer}
Let $f$ be a meromorphic function in
$\bC^n$ of finite lower order $\lambda$, i.e., \eqref{lowerorder} holds.
Then
\begin{equation} \label{Paleylan}
\liminf\limits_{r\to +\infty}
\frac{M\bigl(r; \log |f_{\zeta}|\bigr)}{T(r; f)}
\leq P_n(\lambda )\, , \quad \zeta \in S^n,
\end{equation}
where $P_n(\lambda )$ was defined in \eqref{Pnrho},
and this estimate is  best possible.
\end{Theorem}

The estimate $\eqref{Paleylan}$ was new also for entire functions and
was not implied in general by the above result of
B.~Dahlberg for subharmonic functions in $\R^m$, $m\geq 3$.

On the other hand, one can  pose the problem of finding a complete
and best possible analogue of \eqref{Paleylambda} for plurisubharmonic
functions and entire functions of several variables. Such result was
obtained in our paper \cite[Theorem 1]{Khabib99} (1999):

\begin{Theorem}[B.~N.~Khabibullin] \label{Khabpl}
Let $u$ be a plurisubharmonic function in
$\bC^n$ of finite lower order $\lambda$.
Then
\begin{equation} \label{Paleypsh}
\liminf\limits_{r\to +\infty}\frac{M(r; u)}{T(r; u)}\leq P_n(\lambda )
\end{equation}
for $\lambda \leq 1$  and this estimate is  best possible. For $\lambda >1$,
\begin{equation}  \label{Paleymore1}
\liminf\limits_{r\to +\infty}\frac{M(r; u)}{T(r; u)}\leq
P_1 (\rho ) \, \prod_{k=1}^{n-1}b\Bigl(\frac{\rho}{2k}\Bigr)
\leq e^{n-1}P_n(\rho ) ,
\end{equation}
where $b(x)$ was defined in \eqref{Pb}.
\end{Theorem}

More can be said about the definitive character of \eqref{Paleypsh}.
For each $\rho \geq 0$ there exist in $\bC^n$ entire functions of order
$\rho$ and normal type such that
$$
\lim_{r\to +\infty} \frac{M\bigl( r; \log |f|\bigr)}{T\bigl( r; \log
|f|\bigr)} = P_n(\rho ) \, .
$$

Comparing with the Dahlberg's Theorem  \ref{Dahl}, we can
draw the following  conclusions:
\begin{enumerate}[(a)]
\item  for $\lambda =0$ or $\lambda =1$ the result of the Theorem
\ref{Khabpl} is a consequence of Dahlberg's Theorem;
\item for $0<\lambda <1$ and $n=2$ Theorem \ref{Khabpl} does not follow
from  Dahlberg's Theorem;
\item for $\lambda \geq \pi e-1$ and $n=2$
even the estimate \eqref{Paleymore1} in Theorem \ref{Khabpl} does not
follow from Dahlberg's Theorem;
\item as $\lambda \to +\infty$, for $n=2$ the estimate \eqref{Paleymore1}
is better by order $O(1/\lambda )$ than the one following from
Dahlberg's Theorem.
\end{enumerate}

\subsubsection*{Unsolved problems.}

\paragraph*{Problem 1.}
{\it Extend the results of the Theorems \ref{DDL}--\ref{BBm}
for entire and meromorphic functions of several variables of infinite order.}


\paragraph*{Problem 2.}
{\it Extend the results of the Theorems
\ref{Khabmer}--\ref{Khabpl} to
the Nevanlinna {\rm $\mathbf{\mbox{\L}}$}-characteristic
and to the the Nevanlinna-Jensen {\rm $(\mathbf{\mbox{\L}},
\boldsymbol \mu )$}-characteristic\/}.

\paragraph*{Problem 3.}
{\it Extend the results of the  Theorem \ref{Khabpl} for
the $m_q(r; \cdot)$ instead of $M(r; \cdot )$.}

\paragraph*{Problem 4.}
{\it  Prove  the upper estimate in \eqref{Paleypsh} for all $\lambda$\/}.

\subparagraph*{Commentary.} In order to  prove the upper estimates
in \eqref{Psh} for all $\rho$ and in \eqref{Paleypsh}
for all $\lambda$, it is sufficient  to confirm the following:
\begin{Hypothesis}
Let $S$ be a  nonnegative increasing function on $\R_+$,
$S(0)=0$, and the function $S(t)$ is convex with respect to
$\log t$, i.e., $S(e^x )$ is convex on $[-\infty , +\infty )$.
Further, let $\lambda \geq 1/2$,  $n\in \N$, $n\geq 2$.
If
\begin{equation} \label{Hyppremise}
\int\limits_0^1 S(tx)(1-x^2)^{n-2}x\, dx \leq t^{\lambda}\, ,
\quad 0\leq t < +\infty \, ,
\end{equation}
then
\begin{equation} \label{Hypconcl}
\int\limits_0^{+\infty}
S(t)\frac{t^{2\lambda-1}}{(1+t^{2\lambda})^2}\, dt\leq \frac{\pi
(n-1)}{2\lambda}\prod_{k=1}^{n-1} \Bigl(1+\frac{\lambda}{2k}\Bigr)\, .
\end{equation}
\end{Hypothesis}

This Hypotesis is true if $\lambda \leq 1$. When
$$
S(t)=2(n-1)\prod_{k=1}^{n-1} \Bigl(1+\frac{\lambda}{2k}\Bigr)
\, t^{\lambda}\, , \quad \lambda \geq \frac12 \, ,
$$
we have the equalities in  \eqref{Hyppremise}  and in \eqref{Hypconcl}.

It is not known even, whether the formulated Hypothesis
is true for $n=2$ and $\lambda >1$ ?

%%\newpage

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\bigskip\par\noindent{\slshape
Department of Math., Bashkir State University,
Frunze St., 32, Ufa, Bashkortostan, 450074, Russia}

\bigskip\par\noindent{\slshape
Institute  of Math. of Ufa Scientific Center, Chernishevski\u{\i} St., 112,
Ufa, Bashkortostan, 450025, Russia}

\bigskip\par\noindent {\slshape E-mail: algeom$\, @\,$bsu.bashedu.ru
\hskip0.5cm and(or) \hskip0.5cm khabib-bulat$\, @\,$mail.ru}


\end{document}

math0502433.extracted_bib
\begin{thebibliography}{Kh13}

\bibitem[A]{Akhiezer} N.~I.~Akhiezer
\newblock{\it Novy\u{\i} vyvid neobkhidnykh umov prynalezhnosti}
\newblock{\it tsilo\"{\i} functsi\"{\i} tsilogo por\t{ia}dku do pevnogo
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\newblock{Kiev, Zapiski phys.-matem. otd. AN USSR, V.~2, 1927, No.~3,
P.~29--33 (Ukrainian).}
\newblock{\it A new deduction of necessary conditions
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\bibitem[Beck]{Beck} W.~Beck,
\newblock{\it Efficient quotient representations of meromorphic
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\newblock{Ph.D., Thesis, University of Illinois, Urbana, IL, 1970.}

\bibitem[Be1]{Bergweiler} W.~Bergweiler,
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\bibitem[Be2]{Berg} W.~Bergweiler,
\newblock{\it A question of Gol'dberg concerning entire functions with
prescribed zeros,}
\newblock{J. d'Analyse Math. V.~63, 1994, P.~121--122.}

\bibitem[BB]{BB} W.~Bergweiler and H.~Bock,
\newblock{\it On the growth of meromorphic functions of infinite order,}
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\bibitem[Ch]{Chuang} C.~T.~Chuang,
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\end{thebibliography}