Radius of Close-to-convexity of Harmonic Functions

TL;DR

This paper determines the radius within which harmonic functions close to convex are univalent and starlike, introduces new coefficient inequalities, and improves the Bloch-Landau constant for bounded harmonic mappings.

Contribution

It establishes the radius of close-to-convexity for harmonic functions and introduces new coefficient inequalities that enhance known bounds like the Bloch-Landau constant.

Findings

Radius of univalence is approximately 0.113.

Radius of starlikeness matches the univalence radius.

New coefficient inequality improves the Bloch-Landau constant.

Abstract

Let ${\mathcal H}$ denote the class of all normalized complex-valued harmonic functions $f=h+\bar{g}$ in the unit disk ${\mathbb D}$, and let $K=H+\bar{G}$ denote the harmonic Koebe function. Let $a_n,b_n, A_n, B_n$ denote the Maclaurin coefficients of $h,g,H,G$, and $${\mathcal F}=\{f=h+\bar{g}\in {\mathcal H}:\,|a_n|\leq A_n and |b_n|\leq B_n for n\geq 1}. $$ We show that the radius of univalence of the family ${\mathcal F}$ is $0.112903...$. We also show that this number is also the radius of the starlikeness of ${\mathcal F}$. Analogous results are proved for a subclass of the class of harmonic convex functions in ${\mathcal H}$. These results are obtained as a consequence of a new coefficient inequality for certain class of harmonic close-to-convex functions. Surprisingly, the new coefficient condition helps to improve Bloch-Landau constant for bounded harmonic mappings.