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

Abstract

Let A be tile class of all analytic functions in the unit disk U such that f(0)=f′(0)−1=0. A function f∈A is called starlike with respect to 2n symmetric-conjugate points if Rezf′(z)/fn(z)>0 for z∈U, where fn(z)=12n∑k=0n−1[ω−kf(ωkz)+ωkf(ωkz˜)¯], ω=exp(2πi/n]. This class is denoted by Sn*, and was studied in [1]. A sufficient condition for starlikeness with respect to symmetric-conjugate points is obtained. In addition, images of some subclasses of Sn* under the integral operator I:A→A, I(f)=F where F(z)=c+1(g(z))c∫0zf(t)(g(t))c−1g′(t)dt,   c>0 and g∈A is given are determined.