On Koebe radius and coefficients estimate for univalent harmonic mappings

TL;DR

The paper extends the Koebe-radius problem to broad classes of univalent harmonic mappings by introducing $\mathcal{S}_H^0(k,m)$ with dilatation control $|w_f(z)|\le k|z|^m$. It proves a sharp lower bound $|F(z)|\ge \frac{|z|}{4(1+k|z|^m)^{2/m}}$, implying a disk of radius $\frac{1}{4(1+k)^{2/m}}$ is contained in $F(\mathbb{D})$. Corollaries include explicit Koebe-radius improvements for certain $(k,m)$, coefficient bounds for the holomorphic part, and an area-minimization statement with extremal function $f(z)=z+\frac{k}{m+1}\overline{z}^{m+1}$. The approach relies on extremal-length methods and shearing-type constructions to relate harmonic maps to the classical Koebe map, yielding concrete, verifiable bounds and extending Koebe-type phenomena to harmonic mappings.

Abstract

The problem on estimate of the Koebe radius for univalent harmonic mappings of the unit disk $\mathbb D=\{z\in\mathbb C : |z|<1\}$ is considered. For a subclass of harmonic mappings with the standard normalization and a certain growth estimate for analytic dilatation, we provide new estimate for the Koebe radius. New estimate for Taylor coefficients of the holomorphic part of a function from the subclass under consideration is obtained as a corollary.

Theorems & Definitions (12)

• Theorem 1
• Corollary 1
• Proposition 1
• Proposition 2
• Corollary 2
• Corollary 3
• Proposition 3
• Proposition 4
• proof : Proof of Theorem \ref{['th1']}
• proof : Proof of Proposition \ref{['prop2']}