Argument of bounded analytic functions and Frostman's type conditions

TL;DR

This paper investigates the argument growth of bounded analytic functions in the unit disk using Grishin's measure, providing conditions for the local behavior of their logarithm and zero concentration.

Contribution

It introduces new characterizations of the argument growth and zero distribution of bounded analytic functions based on Frostman's type conditions.

Findings

Established growth estimates for the argument of bounded analytic functions.

Provided necessary and sufficient conditions for the local behavior of the logarithm of such functions.

Linked zero concentration with the argument growth and logarithmic behavior.

Abstract

We describe the growth of the naturally defined argument of a bounded analytic function in the unit disk in terms of the complete measure introduced by A.Grishin. As a consequence, we characterize the local behavior of a logarithm of an analytic function. We also find necessary and sufficient conditions for closeness of $\log f(z)$, $f\in H^\infty$, and the local concentration of the zeros of $f$.