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

摘要

Let H denote the class of all normalized complex-valued harmonic functions f = h + g-bar in the unit disk D, and let S_H~0 denote the class of univalent and sense-preserving functions f in H such that f_z (0) = 0. If K = H + G-bar denotes the harmonic Koebe function whose dilation is ω(z) = z, then K ∈ S0H and it is conjectured that K(z) is extremal for the coefficient problem in S0H. If the conjecture were true, then F contains the family S_H~0, where F = { f = h + g-bar ∈ H: |a_n| ≤ A_n and |b_n| ≤ B_n for n ≥ 1}. Here, an, bn, An, and Bn denote the Maclaurin coefficients of h, g, H, and G.We show that the radius of univalence of the family F is 0.112903....We also show that this number is also the radius of the fully starlikeness of F.Analogous results are proved for a family which contains the class of harmonic convex functions in H. We use the new coefficient estimate for bounded harmonic mappings and Lemma 1.6 to improve Bloch-Landau constant for bounded harmonic mappings.