# Isoperimetric inequalities for Bergman analytic content

## TL;DR

This paper explores isoperimetric inequalities related to the Bergman p-analytic content of planar domains, linking it to torsional rigidity and characterizing cases of equality.

## Contribution

It establishes new isoperimetric inequalities for Bergman p-analytic content and analyzes conditions for equality with bounds.

## Key findings

- Derived inequalities connecting Bergman p-analytic content and torsional rigidity.
- Characterized domains where equality holds in these inequalities.
- Extended concepts to higher dimensions using harmonic vector fields.

## Abstract

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $\Omega $ measures the $L^{p}(\Omega )$-distance between $\overline{z}$ and the Bergman space $A^{p}(\Omega )$ of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman $p$-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.