Sharp coefficient, logarithmic and Schwarzian estimates for $\alpha$-convex functions on the real line

Abstract

The class $M_{\alpha }$ of $\alpha$-convex functions, introduced by Mocanu in 1969, interpolates between starlike ($\alpha =0$) and convex ($\alpha =1$) mappings and has traditionally been studied only for $\alpha \ge 0$. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes $M_{\alpha }$ for every real $\alpha \ne 0$ through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of $\beta$ for which $M_{\alpha }\subset C_{\beta }$ when $\alpha \ge 1$; we also show that the reciprocal inclusion $C_{\beta }\subset M_{\alpha }$ fails for every $\alpha >1$. Building on this, we obtain a sharp Fekete--Szegő inequality, valid for every $\alpha \notin \{0,-1,-1/2\}$, and sharp bounds for the first two logarithmic coefficients of $M_{\alpha }$, which recover the classical values for starlike and convex functions as special cases. We also determine the order of $M_{\alpha }$, establish a piecewise sharp bound for its Schwarzian norm together with an explicit quasiconformal extension criterion via the Ahlfors--Weill theorem, and compute the Schwarzian norm of the $\alpha$-Koebe function explicitly for $\alpha =1/n$, $n\in \mathbb{N}$, verified for $n\le 9$ and numerically supported for $n<50$, leading to a conjecture for the general case. Finally, we propose a conjectural sharp bound for the second Hankel determinant of $M_{\alpha }$, consistent with the known values $1$ and $1/8$ for the starlike and convex subclasses.