arXiv2026
The class $M_α$ of $α$-convex functions, introduced by Mocanu in 1969, interpolates between starlike ($α=0$) and convex ($α=1$) mappings and has traditionally been studied only for $α\geq0$. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes $M_α$ for every real $α\neq0$ through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of $β$ for which $M_α\subset C_β$ when $α\geq1$; we also show that the reciprocal inclusion $C_β\subset M_α$ fails for every $α>1$. Building on this, we obtain a sharp Fekete--Szegő inequality, valid for every $α\notin\{0,-1,-1/2\}$, and sharp bounds for the first two logarithmic coefficients of $M_α$, which recover the classical values for starlike and convex functions as special cases. We also determine the order of $M_α$, 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 $α$-Koebe function explicitly for $α=1/n$, $n\in\mathbb N$, verified for $n\leq9$ 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 $M_α$, consistent with the known values $1$ and $1/8$ for the starlike and convex subclasses.