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 $n \leq 9$ and conjecture to hold in general.
From: Rodrigo Hernández [view email]
[v1]
Fri, 19 Jun 2026 16:15:47 UTC (151 KB)
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