Some properties of starlike functions with respect to symmetric-conjugate points

Abstract

LetAbe tile class of all analytic functions in the unit diskUsuch thatf(0)=f′(0)−1=0. A functionf∈Ais called starlike with respect to2nsymmetric-conjugate points ifRezf′(z)/fn(z)>0forz∈U, wherefn(z)=12n∑k=0n−1[ω−kf(ωkz)+ωkf(ωkz˜)¯],ω=exp(2πi/n]. This class is denoted bySn*, and was studied in [1]. A sufficient condition for starlikeness with respect to symmetric-conjugate points is obtained. In addition, images of some subclasses ofSn*under the integral operatorI:A→A,I(f)=FwhereF(z)=c+1(g(z))c∫0zf(t)(g(t))c−1g′(t)dt,   c>0andg∈Ais given are determined.