Article
Ufa Mathematical Journal
Volume 3, Number 1, pp. 92-100
On the growth of the maximum modulus of an entire function depending on the growth of its central index.
Filevych P.V.
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Let $h$ be a positive continuously function on $(0,+\infty)$, $f(z)=\sum_{n=0}^\infty a_nz^n$ be an entire function, and $M_f(r)=\max\{|f(z)|:|z|=r\}$, $\mu_f(r)=\max\{|a_n|r^n:n\ge 0\}$ and $\nu_f(r)=\max\{n\ge 0: |a_n|r^n=\mu_f(r)\}$ be the maximum modulus, the maximal term, and the central index of the function f, respectively. We establish necessary and sufficient conditions on the growth of $\nu_f(r)$ under which $M_f(r)=O(\mu_f(r) h(\ln\mu_f(r)))$, $r\to+\infty$.