| Abstract |
Let $\mathcal{L}$ be the class of positive continuous functions on
$(-\infty,+\infty)$ and let $\mathcal{L}_+^2$ be the class of positive
continuous increasing with respect to each variable functions
$\gamma$ in $\mathbb{R}^2$ such that $ \gamma(r_1,r_2)\to +\infty
$ as $r_1+r_2\to+\infty.$ We prove the following} statement:
for all entire functions of the form $
f(z_1,z_2)=\sum_{n+m=0}^{+\infty}a_{nm}z_1^nz_2^m$ such that $
|a_{nm}|\leq\exp\{-(n+m)\psi(n,m)\} \mbox{ for } n+m\geq k_0(f)$ and functions $f(z_1,1), f(1,z_2)$ are transcendent,
$ \psi\in\mathcal{L}_+^2, $ the inequality
$$
\mathfrak{M}_f(r_1,r_2)=O(M_f(r_1,r_2) h(\ln M_f(r_1,r_2))),\
h\in\mathcal{L},\
r^{\vee}=\min\{r_1,r_2\}\to+\infty,
$$
holds where $M_f(r_1,r_2)=\max\{|f(z_1,z_2)|\colon
|z_1|=r_1,|z_2|=r_2\},$\
$\mathfrak{M}_f(r_1,r_2)=\sum_{n+m=0}^{+\infty}|a_{nm}|\times$ $\times r_1^nr_2^m,$
if and only if
\begin{equation*}
(\forall\gamma\in\mathcal{L}_+^2)\colon\ \sqrt{r_1r_2}=O\big(h(\gamma(r_1,r_2)\psi(r_1,r_2))\big), \
r^{\vee}\to+\infty.
\end{equation*} |
| Reference |
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