Bitlyan-Gol'dberg type inequality for entire functions and diagonal maximal term

Abstract

In the article is obtained an analogue of Wiman-Bitlyan-Gol'dberg type inequality for entire $f\colon\mathbb{C}^p\to \mathbb{C}$ from the class $\mathcal{E}^{p}(\lambda)$ of functions represented by gap power series of the form
$$
f(z)=\sum\limits_{k=0}^{+\infty} P_k(z),\quad
z\in\mathbb{C}^p.
$$
Here $P_0(z)\equiv a_{0}\in\mathbb{C},$ $
P_k(z)=\sum_{\|n\|=\lambda_k} a_{n}z^{n}$ is homogeneous
polynomial of degree $\lambda_k\in\mathbb{Z}_+,$ ànd $ 0=\lambda_0<
\lambda_k\uparrow +\infty$\ $(1\leq k\uparrow +\infty ),$
$\lambda=(\lambda_k)$.\ We consider the exhaustion of the
space\ $\mathbb{C}^{p}$\
by the system $(\mathbf{G}_{r})_{r\geq 0}$ of a bounded complete multiple-circular domains $\mathbf{G}_{r}$
with the center at the point $\mathbf{0}=(0,\ldots,0)\in \mathbb{C}^{p}$. Define $M(r,f)=\max\{|f(z)|\colon z\in\overline{G}_r\}$, $\mu(r,f)=\max\{|P_k(z))|\colon z\in\overline{G}_r\}$.
Let $\mathcal{L}$ be the class of positive continuous functions $\psi\colon \mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $\int_{0}^{+\infty}\frac{dx}{\psi(x)}<+\infty$, $n(t)=\sum_{\lambda_k\leq t}1$ counting function of the sequence $(\lambda_k)$ for $t\geq 0$. The following statement is proved:
{\it If a sequence $\lambda=(\lambda_{k})$ satisfy the condition
\begin{equation*}
(\exists p_1\in (0,+\infty))(\exists t_0>0)(\forall t\geq t_0)\colon\quad n(t+\sqrt{\psi(t)})-n(t-\sqrt{\psi(t)})\leq t^{p_1}
\end{equation*}
for some function $\psi\in \mathcal{L}$,
then for every entire function $f\in\mathcal{E}^{p}(\lambda)$, $p\geq 2$ and for any
$\varepsilon>0$ there exist a constant $C=C(\varepsilon, f)>0$ and a set $E=E(\varepsilon, f)\subset [1,
+\infty)$ of finite logarithmic measure such that the inequality
\begin{equation*}
M(r, f)\leq C m(r,
f)(\ln m(r, f))^{p_1}(\ln\ln m(r, f))^{p_1+\varepsilon}
\end{equation*}
holds for all $ r\in[1,
+\infty]\setminus E$.}

The obtained inequality is sharp in general.
At $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and the Bitlyan-Gol'dberg inequality (1959) it follows. In the case $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and from obtained statement we get the assertion on the Bitlyan-Gol'dberg inequality (1959), and at $p=1$ about the classical Wiman inequality it follows.