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

TL;DR

This paper investigates the improvement of Wiman-Valiron inequalities for classes of analytic functions in the unit disk, using Baire category methods and probabilistic approaches to identify almost sure bounds.

Contribution

It introduces new conditions on the sequence of angular frequencies to almost surely enhance Wiman-Valiron inequalities for analytic functions.

Findings

Almost sure improvement of Wiman-Valiron inequalities under specified conditions.

Identification of function classes where inequalities can be strengthened.

Establishment of measure-theoretic properties related to the improvements.

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 $r\in[0, 1).$