Baire categories and classes of analytic functions in which the Wiman-Valiron type inequality can be almost surely improved

Abstract

Let $f(z)={\sum }_{n=0}^{+\infty }{a}_n{z}^n$\ $(z\in \mathbb{C})$\ be an analytic function in the unit disk and $f_t$ be an analytic function of the form $f_t(z)={\sum }_{n=0}^{+\infty }{a}_n{e}^{i{\theta }_{n}t}{z}^n,$ where $t\in \mathbb{R},$ ${\theta }_n\in \mathbb{N},$ and $h$ be a positive continuous function on $(0,1)$ increasing to $+\infty$ and such that ${\int }_{r_0}^{1}h(r)dr=+\infty ,r_0\in (0,1).$\ If the sequence $({\theta }_n{)}_{n\ge 0}$ satisfies the inequality $$\underset{n\to +\infty }{\overline{\lim }}\frac{1}{\ln n}\ln \frac{{\theta }_n}{{\theta }_{n+1}-{\theta }_n}\le \delta \in [0,1/2),$$ then for all analytic functions $f_t$ almost surely for $t$ there exists a set $E=E(\delta ,t)\subset (0,1)$ such that ${\int }_{E}h(r)dr<+\infty$ and $$\underset{substackr\to 1-0r\notin Esubstack}{\overline{\lim }}\frac{\ln M_f(r,t)-\ln {\mu }_f(r)}{2\ln h(r)+\ln \ln \{h(r){\mu }_f(r)\}}\le \frac{1+2\delta }{4+3\delta },$$ where $M_f(r,t)=\max \{|f_t(z)|:|z|=r\},$\ ${\mu }_f(r)=\max \{|a_n|r^n:n\ge 0\}$\ for $r\in [0,1).$

17 pages