A subclass of harmonic univalent functions with negative coefficients

Abstract

Complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form f=h+ḡ, where h and g are analytic in U. In this paper, consider the class HP(β), (0⩽β<1) consisting of harmonic and univalent functions f=h+ḡ for which Re{h′(z)+g′(z)}>β. We give sufficient coefficient conditions for normalized harmonic functions in the class HP(β). These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.