A Characterization of α-Convex Functions with Sharp Coefficient and Schwarzian Estimates

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--Szeg\H{o} 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 $n \leq 9$ and conjecture to hold in general.