Notes on the starlikeness of an integral transform

Abstract

Let $ A $ denote the class of normalized analytic functions in the unit disk $ D $. For $ f(z) \in A $ and $ \alpha > 0 $, let $ F_{\alpha} (z) = \int_0^z (f(t)/t)^{\alpha} dt $ in $ D $. In this note, the author obtains the best constant $ \beta (\alpha) $ for each $ \alpha \in (0,3] $ such that Re$ \{ f'(z) (f(z)/z)^{\alpha -1} \} > \beta (\alpha) $ implies the starlikeness of $ F_{\alpha} (z) $.