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

Abstract

Let H denote the class of all normalized complex-valued harmonic functions f = h + g in the unit disk D, and let S 0 H denote the class of univalent and sense-preserving functions f in H such that f z (0) = 0. If K = H + G denotes the harmonic Koebe function whose dilation isHere, a n , b n , A n , and B n 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.

Harmonic close-to-convex functions and minimal surfaces

“…In 1980, Mocanu [10] (see also [13]) proved that if f = h+g is a harmonic mapping in a convex domain Ω such that Re (h (z)) > |g (z)| for all z ∈ Ω, then f is univalent and sense-preserving in Ω. An improved version of this results was given in [8,13]. In order to discuss a general situation, it is appropriate to recall the following result due to Mocanu [10].…”

On a subclass of close-to-convex harmonic mappings

“…In [13], Chen et al obtained some results of the class M(α, −1/2) with |α| = 1. We refer to [22][23][24][25][26][27][28] for discussions on close-to-convex harmonic mappings.…”

Harmonic Mappings Related to Functions With Bounded Boundary Rotation and Norm of the Pre-Schwarzian Derivative

“…The reasoning used in a proof of Theorem 2.1 may be applied to the bounds of coefficients of any harmonic functions f = h +ḡ, with an assumption |g ′ (0)| = α and such that the coefficients of the analytic part h satisfy |a n | ≤ B n for n ≥ 1 (here h(z) = z + a 2 z 2 + · · · ). Such approach is also presented in [12].…”