Construction of Subclasses of Univalent Harmonic Mappings

Abstract

Abstract. Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk are widely studied. A new methodology is employed to construct subclasses of univalent harmonic mappings from a given subfamily of univalent analytic functions. The notions of harmonic Alexander operator and harmonic Libera operator are introduced and their properties are investigated.

On a subclass of harmonic close-to-convex mappings

“…In 1982, R. Singh and S. Singh [28] proved that W(1) is a subclass of analytic starlike functions. In 2014, Nagpal and Ravichandran [17] studied the following class…”

“…H is a subclass of S * H 0 and P 0 H . In particular, the members of W 0 H are fully starlike in D. The sharp coefficient bounds and the growth theorem for functions in the class W 0 H have been investigated in [17]. It has been proved that the class W 0 H is closed under convolution and convex combinations.…”

Univalence and convexity in one direction of the convolution of harmonic mappings

“…In [9], the authors introduced the notion of positive harmonic Alexander operator Λ + H : H → H defined by…”

Close-to-convexity of a class of harmonic mappings defined by a third-orderdifferential inequality

“…Recently, Ghosh and Vasudevarao [5] defined a class of functions f = h +ḡ ∈ H 0 satisfying the condition Re {h (z) + αzh (z)} > |g (z) + αzg (z)| for z ∈ U and they investigated coefficient bounds, growth estimates, convolution and radius of convexity for the partial sums of members of their class. Other interesting studies of harmonic mappings that we have inspired in this work are [2,9,10,14,16].…”