On a subclass of harmonic close-to-convex mappings

Abstract

Let H denote the class of harmonic functions f in D := {z ∈ C : |z| < 1} normalized by f (0) = 0 = f z (0) − 1. For α ≥ 0, we consider the following classIn this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class W 0 H (α). We also prove growth theorem, convolution, convex combination properties for functions in the class W 0 H (α). Finally, we determine the value of r so that the partial sums of functions in the class W 0 H (α) are closeto-convex in |z| < r.1991 Mathematics Subject Classification. Primary 30C45, 30C80.

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

“…Remark 12 (i) Our results, obtained in this section, yield the results of the classes R 0 H (0, 0) and R 0 H (λ, 0) which are defined and studied in [12,13,15] and [5], respectively.…”

“…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].…”

New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality

“…In all studies mentioned above [10,17,18,23], it is proved that the functions in corresponding classes are close-to-convex. Also, coefficient bounds, growth estimates, and convolution properties of the classes are obtained.…”

“…Ghosh and Vasudevarao [10] investigated radius of convexity for the partial sums of members of the class…”

New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality

“…H and in particular, the members of the class are fully starlike in U. Ghosh and Vasudevarao [8] investigated radius of convexity for the partial sums of members of the class…”