Bergman-H\"older Functions, Area Integral Means and Extremal Problems

TL;DR

This paper investigates weighted area integral means of analytic functions in the unit disc, linking their growth to mean H"older continuity in Bergman spaces, and introduces second iterated difference quotients for new insights.

Contribution

It introduces the use of second iterated difference quotients to analyze growth of integral means and applies findings to extremal problems in Bergman spaces.

Findings

Growth of integral means relates to mean H"older continuity.

Second iterated difference quotients provide new analytical tools.

Applications to extremal problems in Bergman spaces.

Abstract

We study certain weighted area integral means of analytic functions in the unit disc. We relate the growth of these means to the property of being mean H\"older continuous with respect to the Bergman space norm. In contrast with earlier work, we use the second iterated difference quotient instead of the first. We then give applications to Bergman space extremal problems.