Some properties of starlike functions with respect to symmetric-conjugate points (Q1895270)

Summary: Let \(A\) be the class of all analytic functions in the unit disk \(U\) such that \(f(0)=f'(0)-1=0\). A function \(f\in A\) is called starlike with respect to \(2n\) symmetric-conjugate points if \(\Re zf'(z)/f_ n(z)>0\) for \(z\in U\), where \[ f_ n(z)=\frac{1}{2n}\sum^ {n-1}_ {k=0}[\omega^ {-k}f(\omega^ kz)+\omega^ k \overline {f(\omega^ k\overline z)}], \] \(\omega=\exp (2\pi i/n)\). This class is denoted by \(S^ *_ n\) and was studied in [\textit{H. S. Al-Amiri, D. Coman} and \textit{P. T. Mocanu}, Glas. Mat., III. Ser. 30, No. 2, 209--219 (1995; Zbl 0968.30009)]. A sufficient condition for starlikeness with respect to symmetric-conjugate points is obtained. In addition, images of some subclasses of \(S^ *_ n\) under the integral operator \(I\colon A\to A\), \(I(f)=F\), where \[ F(z)=\frac{c+1}{(g(z))^ c}\int^ z_ 0f(t)(g(t))^ {c-1}g'(t)\,dt,\quad c>0 \] and \(g\in A\) is given, are determined.