Notes on the starlikeness of an integral transform (Q2752336)

Let \(A\) denote the class of all normalized analytic functions in the unit disc \(\Delta =\{z: |z|<1\}\). For \(f\in A\), let \(F_\alpha (z)=\int _0^z(f(t)/t)^{\alpha} dt\). In this note the author finds conditions on \(\alpha \) and \(\beta (\alpha) \) so that \(\text{Re }\{f'(z)(f(z)/z)^{\alpha -1}\}>\beta (\alpha)\) implies that \(F_{\alpha }(z)\) is starlike in \(\Delta\). A similar problem has been considered by a number of authors.