Radius of Ma-Minda starlikeness of certain normalised analytic functions

Abstract

We find the radius of Ma-Minda starlikeness  of normalised analytic functions of the form $g(z)=z(f'(z))^\alpha$,  $\alpha > 0$ where $f$ is in the class $ \mathcal{CV}[A,B]$ of Janowski convex functions and $g(z)=z(zf'(z)/f(z))^\alpha$,  $\alpha > 0$ where $f$ is in the class $\mathcal{CV}$ of convex functions. As particular cases, we obtain criteria for these  functions to belong to certain Ma-Minda classes.