On directional convexity of harmonic mappings in the plane

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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 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.