On directional convexity of harmonic mappings in the plane

Abstract

Let denote the class of all complex-valued harmonic functions f in the open unit disk normalized by f(0) = f_z(0) − 1 = = 0, and 𝒜 the subclasses of consisting of univalent and sense-preserving functions and normalized analytic functions, respectively. For φ ∈ 𝒜, let := {f = h + ḡ ∈ : h – e^2αi g = φ} be subfamily of . In this paper, we shall determine the conditions under which the analytic function φ with φ ∈ 𝒜, the linear convex combination tf₁ + (1 − t)f₂ with f_j ∈ , j = 1, 2, and the harmonic convolution f₁ ∗ f₂ with f_j ∈ , j = 1, 2, are univalent and convex in one direction, respectively. Many previous related results are generalized.

Mathematics Subject Classification (2010):

• 31A05
• 30C45

Key words:

• Harmonic mappings
• convex in one direction
• convolution
• linear convex combination

Notes

1 Supported by NSFC (11501002), Natural Science Foundation of Anhui Province (1908085MA18), Foundations of Anhui Educational Committee (KJ2017A029) and Anhui Uni- versity (Y01002428), China.