Partial Sums of Starlike Harmonic Univalent Functions

Abstract

Abstract. Although, interesting properties on the partial sums of analytic univalentfunctions have been investigated extensively by several researchers, yet analogous resultson partial sums of harmonic univalent functions have not been so far explored. The mainpurpose of the present paper is to establish some new and interesting results on the ratioof starlike harmonic univalent function to its sequences of partial sums. 1. IntroductionA continuous complex-valued function f = u+ ivis said to be harmonic ina simply connected domain D if both uand v are real harmonic in D. In anysimply connected domain we can writef = h+ g, where hand gare analytic inD. We call hthe analytic part and gthe co-analytic part of f:A necessary andsucient condition forf to be locally univalent and sense-preserving in Dis thatjh 0 (z)j>jg(z)j;z2D:See Clunie and Sheil-Small [2].Denote by S H the class of functions f= h+gwhich are harmonic univalent andsense-preserving in the unit disk U= fz: jzj<1gfor which f(0) = f