Research Article

On directional convexity of harmonic mappings in the plane

Published in: Quaestiones Mathematicae
Volume 43, issue 10, 2020 , pages: 1435–1447
DOI: 10.2989/16073606.2019.1632955
Author(s): Bo-Yong Long, China, Hua-Ying Huang, China

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 ???? the subclasses of consisting of univalent and sense-preserving functions and normalized analytic functions, respectively. For φ????, let := {f = h + : he 2αi g = φ} be subfamily of . In this paper, we shall determine the conditions under which the analytic function φ with φ????, the linear convex combination tf 1 + (1 − t)f 2 with fj , j = 1, 2, and the harmonic convolution f 1f 2 with fj , j = 1, 2, are univalent and convex in one direction, respectively. Many previous related results are generalized.

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