A Characterization of $\alpha$-Convex Functions with Sharp Coefficient and Schwarzian Estimates

Abstract

The class $M_{\alpha }$ of $\alpha$-convex functions, introduced by Mocanu in 1969, interpolates between starlike and convex functions. We prove a characterization of $M_{\alpha }$ that extends a theorem of Chuaqui, Duren, and Osgood from the convex case to the full class, and determine sharp values of $\beta$ for which $M_{\alpha }\subset C_{\beta }$ and $C_{\beta }\subset M_{\alpha }$. We also obtain a sharp Fekete--Szegő inequality, bounds for the order and the Schwarzian norm, and an explicit formula for the Schwarzian norm of the $\alpha$-Koebe function for $\alpha =1/n$, $n\in \mathbb{N}$, which we verify for $n\le 9$ and conjecture to hold in general.