Starlikeness, convexity and close-to-convexity of harmonic mappings

TL;DR

This paper extends classical results on harmonic mappings by exploring conditions for starlikeness and convexity, analyzing properties of a specific harmonic function class, and providing coefficient estimates, growth results, and sharp bounds.

Contribution

It generalizes univalence and convexity conditions for harmonic mappings and investigates the properties of the class \mathcal{M}(\alpha), including sharp bounds.

Findings

Extended conditions for fully starlike and convex harmonic mappings.

Derived coefficient estimates and growth results for \mathcal{M}(\alpha).

Established sharp bounds for the radius of convexity.

Abstract

In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investgate the properties of functions in the class $\mathcal{M}(\alpha)$ $(|\alpha|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=\alpha zh'(z)$, $\RE (1+{zh''(z)}/{h'(z)})>-{1}/{2} $ $ \mbox{for} |z|<1 $. The coefficient estimates, growth results, area theorem and bounds for the radius of starlikeness and convexity of the class $\mathcal{M}(\alpha)$ are determined. In particular, the bound for the radius of convexity is sharp for the class $\mathcal{M}(1)$.