Radius of Close-to-convexity of Harmonic Functions

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 $\mathrm{0.112903...}$. 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.

13 pages