Starlikeness, Convexity and Close-to-convexity of Harmonic Mappings

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 chapter, 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 investigate the geometric properties of functions in the class ${\cal M}(\alpha)$ $(|\alpha|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=\alpha zh'(z)$, Re $(1+{zh''(z)}/{h'(z)})>-{1}/{2}$ for $|z|<1$. The coefficient estimates, growth results, area theorem and bounds for the radius of starlikeness, and convexity of the class ${\cal M}(\alpha)$ are determined. In particular, the bound for the radius of convexity is sharp for the class ${\cal M}(1)$.