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

Abstract

Let $f(z)=\sum_{n=0}^{+\infty} a_nz^n$\ $(z\in\mathbb{C})$\ be an analytic function in the unit disk and $f_t$ be an analytic function of the form $f_t(z)=\sum_{n=0}^{+\infty} a_ne^{i\theta_nt}z^n,$ 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}^1h(r)dr=+\infty, r_0\in(0,1).$\ If the sequence $(\theta_n)_{n\geq0}$ satisfies the inequality $$ \varlimsup_{n\to+\infty}\frac1{\ln n}\ln\frac{\theta_n}{\theta_{n+1}-\theta_n}\leq\delta\in[0,1/2), $$ then for all analytic functions $f_t$ almost surely for $t$ there exists a set $E=E(\delta,t)\subset(0,1)$ such that $\int_Eh(r)dr<+\infty$ and $$ \varlimsup_{{substack} {r\to1-0 r\notin E}{substack}} \frac{\ln M_f(r,t)-\ln\mu_f(r)}{2\ln h(r)+\ln\ln\{h(r)\mu_f(r)\}}\leq\frac{1+2\delta}{4+3\delta}, $$ where $M_f(r,t)=\max\{|f_t(z)|\colon |z|=r\},$\ $\mu_f(r)=\max\{|a_n|r^n\colon n\geq 0\}$\ for $r\in[0, 1).$