Charatheodory and Smirnov type theorem for harmonic mappings

TL;DR

This paper extends classical theorems and inequalities to harmonic homeomorphisms of the unit disk onto Jordan surfaces with rectifiable boundaries, without requiring boundary smoothness.

Contribution

It proves Smirnov and Carathéodory type theorems for harmonic mappings onto Jordan surfaces with rectifiable boundaries, and establishes isoperimetric and Riesz--Zygmund inequalities without boundary smoothness assumptions.

Findings

Proved Smirnov type theorem for harmonic homeomorphisms.

Established Carathéodory type theorem for harmonic mappings.

Derived classical inequalities for Jordan harmonic surfaces without boundary smoothness.

Abstract

We prove a version of Smirnov type theorem and Charatheodory type theorem for a harmonic homeomorphism of the unit disk onto a Jordan surface with rectifiable boundary. Further we establish the classical isoperimetric inequality and Riesz--Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptions of the boundary.