Maximum modulus of entire functions of two variables and arguments of coefficients of double power series

Author O. B. Skaskiv, A. O. Kuryliak,
Lviv Ivan Franko National University

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*}
Keywords entire functions; power series; maximum modulus
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Pages 162-175
Volume 36
Issue 2
Year 2011
Journal Matematychni Studii
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