Research Article

On directional convexity of harmonic mappings in the plane

Published in: Quaestiones Mathematicae
Volume 43 , issue 10 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 1435–1447
DOI: 10.2989/16073606.2019.1632955
Author(s): Bo-Yong Long , China, Hua-Ying Huang , China,
Keywords: 31A05, 30C45, 31A05, 30C45,
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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 + 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 ∈ ????, the linear convex combination tf 1 + (1 − t)f 2 with fj ∈ , j = 1, 2, and the harmonic convolution f 1 ∗ f 2 with fj ∈ , j = 1, 2, are univalent and convex in one direction, respectively. Many previous related results are generalized.
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