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

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.