Sharp bounds for the growth and distortion of the analytic part of convex harmonic functions

TL;DR

This paper establishes precise bounds on how the analytic component of convex harmonic functions can grow and distort, providing fundamental limits for these mappings in complex analysis.

Contribution

It introduces sharp bounds for the growth and distortion of the analytic parts of convex harmonic mappings, advancing understanding of their geometric behavior.

Findings

Sharp upper and lower bounds for growth of the analytic part

Precise distortion limits for convex harmonic mappings

Enhanced understanding of harmonic mapping behavior in convex domains

Abstract

We obtain the sharp upper and lower bounds for the growth and distortion of the analytic parts $h$ of orientation-preserving harmonic mappings $f=h+\overline g$ (normalized in the standard way) that map the unit disk onto a convex domain.