On directional convexity of harmonic mappings in the plane

Abstract

Let ℋ denote the class of all complex-valued harmonic functions ? in the open unit disk normalized by f(0) = f _z(0) − 1 =  = 0,  and ? the subclasses of ℋ consisting of univalent and sense-preserving functions and normalized analytic functions, respectively. For φ ∈ ?, let  := {f = h + ḡ ∈  : h – e ^2αi g = φ} be subfamily of ℋ. In this paper, we shall determine the conditions under which the analytic function ? with ? ∈ ?, the linear convex combination ttf ₁ + (1 − t)f ₂ with f_j ∈ , j = 1, 2, and the harmonic convolution f ₁ ∗ f ₂ with f_j ∈ , j = 1, 2, are univalent and convex in one direction, respectively. Many previous related results are generalized.