Growth, zero distribution and factorization of analytic functions of moderate growth in the unit disc

TL;DR

This paper surveys the zero distribution and factorization of analytic functions of finite order in the unit disc, focusing on growth conditions and boundedness of p-means for Blaschke products.

Contribution

It provides a necessary and sufficient condition for the uniform boundedness of p-means of log-modulus of Blaschke products, advancing understanding of their growth behavior.

Findings

Characterization of zero distribution for finite order functions

Necessary and sufficient conditions for p-means boundedness

Insights into factorization of analytic functions in the unit disc

Abstract

We give a survey of results on zero distribution and factorization of analytic functions in the unit disc in classes defined by the growth of $\log|f(re^{i\theta})|$ in the uniform and integral metrics. We restrict ourself by the case of finite order of growth. For a Blaschke product $B$ we obtain a necessary and sufficient condition for the uniform boundedness of all $p$-means of $\log|B(re^{i\theta})|$, where $p>1$.