Functions starlike with respect to other points (Q1178780)

In this paper some results concerning the generalizations of functions starlike with respect to symmetric points were obtained. If we put \[ S_ s^*(\beta)=\{f(z)=z+\dots:Re{zf'(z)\over f(z)-f(- z)}>\beta, \mid z\mid<1\} \] then the following results (among others) were obtained: If \(f(z)\in S_ s^*(\beta)\) then \[ Re\left( {f(z)-f(-z)\over 2z}\right)^{1/(1-\beta)}\geq {1\over 1+r^ 2}>{1\over 2}, \mid z\mid<1 \] and the function \[ H(z)={a+1\over 2z^ a}\int^ z_ 0 t^{a-1}(f(t)-f(-t))dt \] also belongs to \(S_ s^*(\beta)\) for \(a+\beta>0\). Some other related classes where \(f(-z)\) was replaced by \(\overline{f(\bar z)}\) or by \(\overline{-1(-\bar z)}\) respectively were also investigated.