Inequalities of harmonic univalent functions with connections of hypergeometric functions

Abstract

Let SH be the class of functions f = h + (g) over bar that are harmonic univalent and sense-preserving in the open unit disk U = {z : vertical bar z vertical bar < 1} for which f(0) = f'(0) - 1 = 0. In this paper, we introduce and study a subclass H(alpha, beta)of the class SH and the subclass NH(alpha, beta) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H(alpha, beta) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions.