Weissler and Bernoulli type inequalities in Bergman spaces

TL;DR

This paper investigates Weissler and Bernoulli type inequalities within Bergman spaces with radial weights, establishing conditions on weights for these inequalities to hold and providing counterexamples where they fail.

Contribution

It introduces new conditions on radial weights in Bergman spaces that ensure Weissler type inequalities, and explores analogs of Bernoulli inequalities for noninteger exponents.

Findings

Conditions on weight moments guarantee inequalities for integer exponents.

A special case of the inequality is proved for noninteger exponents.

Counterexample weight shows inequalities do not always hold.

Abstract

We consider Weissler type inequalities for Bergman spaces with general radial weights and give conditions on the weight $w$ in terms of its moments ensuring that $\|f_r\|_{A^{2n}(w)}\leq \|f\|_{A^2(w)}$ whenever $n\in \mathbb{N}$ and $0< r\le 1/\sqrt{n}$. For noninteger exponents a special case of this inequality is proved which can be considered as a certain analog of the Bernoulli inequality. An example of a monotonic weight is constructed for which these inequalities are no longer true.