Abstract
Let denote the class of all complex-valued harmonic functions f in the open unit disk normalized by f(0) = fz(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 – e2αi 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 fj ∈ , j = 1, 2, and the harmonic convolution f1 ∗ f2 with fj ∈ , j = 1, 2, are univalent and convex in one direction, respectively. Many previous related results are generalized.
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.