SOME APPLICATIONS OF FRACTIONAL CALCULUS OPERATORS TO THE ANALYTIC PART OF HARMONIC UNIVALENT FUNCTIONS

Abstract

Recently, Jahangiri [4] studied the harmonic starlike functions of order α, and he defined the class TH(α) consisting of functions f = h + g¯, where h and g are the analytic and the co-analytic part of the function f, respectively. In [3] the author introduced the class TH(α, β) of analytic functions and he proved various coefficient inequalities and growth and distortion theorems, and obtained the radius of convexity for the function h if the function f belongs to the classes TH(α) and TH(α, β). In this paper, we derive various distortion theorems for the fractional calculus and the fractional integral operator of the function h, the analytic part of the function f, if the function f belongs to the class TH(α, β).