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

TL;DR

The paper addresses sharp growth and distortion bounds for the analytic part $h$ of orientation-preserving harmonic maps $f=h+\overline g$ that map the unit disk onto convex domains. By leveraging convexity-in-direction criteria and Carathéodory-type bounds, the author derives final sharp inequalities: $\frac{2|z|+|z|^2}{2(1+|z|)^2}\le |h(z)|\le \frac{2|z|-|z|^2}{2(1-|z|)^2}$ and $|h'(z)|\le \frac{1}{(1-|z|)^3}$ for all $z\in{\mathbb D}$. Equality occurs only for rotations of the harmonic half-plane map $L$, linking the extremal behavior directly to Koebe-function-related structures; a lower bound is obtained via a disk-dilation argument, and sharpness is tied to the Koebe function through a dedicated proposition. These results illuminate the constraints on the analytic part of convex harmonic mappings and provide a concrete, extremal description within the broader class ${\mathcal K}_H^0$. The work also yields corollaries about the image of $h$ containing a disk of radius $3/8$ and strengthens connections to the classical theory of univalent functions.

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.