Radius of close-to-convexity and fully starlikeness of harmonic mappings

Abstract

Let denote the class of all normalized complex-valued harmonic functions in the unit disk , and let denote the class of univalent and sense-preserving functions in such that . If denotes the harmonic Koebe function whose dilation is , then and it is conjectured that is extremal for the coefficient problem in . If the conjecture were true, then contains the family , where

Here, and denote the Maclaurin coefficients of and . We show that the radius of univalence of the family is . We also show that this number is also the radius of the fully starlikeness of . Analogous results are proved for a family which contains the class of harmonic convex functions in . We use the new coefficient estimate for bounded harmonic mappings and Lemma 1.6 to improve Bloch-Landau constant for bounded harmonic mappings.

Keywords:

• coefficient inequality
• partial sums
• radius of univalence
• analytic, univalent, convex and starlike harmonic functions

AMS Subject Classifications:

• Primary: 30C45

Acknowledgments

The authors thank the referee for many helpful comments. The research of the second author was supported by the Academy of Finland, Project No. 2600066611, coordinated by the third author. The original article of the authors from http://arxiv.org/abs/1107.0610 does contain some general results. The second author is currently at Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India.