# Contractive inequalities for Bergman spaces and multiplicative Hankel forms

## TL;DR

This paper establishes sharp inequalities for Bergman spaces, extends them to Dirichlet series, and explores their implications for multiplicative Hankel forms, revealing new relationships and counterexamples in complex analysis.

## Contribution

It introduces novel sharp inequalities for Bergman spaces and connects them to multiplicative Hankel forms for Dirichlet series, including new relationships and counterexamples.

## Key findings

- Derived sharp inequalities for Bergman spaces.
- Extended inequalities to Dirichlet series spaces.
- Identified new relationships between multiplicative Hankel forms.

## Abstract

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc.