Abstract
In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.
Highlights
A complex-valued function f u + iv defined on the unit disk U {z ∈ C: |z| < 1} is called harmonic mapping if u and v are real-valued harmonic functions
For comprehensive and fundamental knowledge on planar harmonic mappings, see Duren [2]. e subclass of S0H denoted by S0CHD consists of all univalent harmonic functions which maps onto domain convex in the direction of the real axis
Dorff and Rolf [17] developed a necessary condition for convex combination to be univalent maps onto a domain convex in the direction of the imaginary axis
Summary
A complex-valued function f u + iv defined on the unit disk U {z ∈ C: |z| < 1} is called harmonic mapping if u and v are real-valued harmonic functions. E subclass of S0H denoted by S0CHD consists of all univalent harmonic functions which maps onto domain convex in the direction of the real axis. Clunie and Sheil [13] proposed an integrated way to construct a univalent harmonic mapping convex in a given direction. Dorff and Rolf [17] developed a necessary condition for convex combination to be univalent maps onto a domain convex in the direction of the imaginary axis. Beig et al [23] studied and found necessary conditions for the convex combination of the right half-plane mappings, the vertical strip mapping, their rotations, and some other harmonic mappings to be univalent and convex in a particular direction.
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