# A sharp inequality for harmonic diffeomorphisms of the unit disk

## TL;DR

This paper extends the Schwarz-Pick inequality to harmonic mappings between the unit disk and Jordan domains with specified perimeter, identifying extremals as harmonic diffeomorphisms solving a second-order Beltrami equation.

## Contribution

It introduces a new inequality for harmonic diffeomorphisms of the unit disk into Jordan domains with fixed perimeter, highlighting extremals as solutions to a specific Beltrami equation.

## Key findings

- Extended Schwarz-Pick inequality for harmonic maps
- Identified extremals as harmonic diffeomorphisms solving a second-order Beltrami equation
- Applicable to convex Jordan domains with given perimeter

## Abstract

We extend the classical Schwarz-Pick inequality to the class of harmonic mappings between the unit disk and a Jordan domain with given perimeter. It is intriguing that the extremals in this case are certain harmonic diffeomorphisms between the unit disk and a convex domain that solve the Beltrami equation of second order.