On spherically convex univalent functions (Q1591629)

A domain \(G\) on the Riemann sphere \(\overline C\) is called spherically convex if, for any pair \(w_1,w_2\in G\), the smaller arc of the greatest circle (spherical geodesic) between \(w_1\) and \(w_2\) also lies in \(G\). A meromorphic univalent function \(f\) in the unit disk \(D\) is called spherically convex \((s\)-convex) if \(f(D)\) is a spherically convex domain in \(\overline C\). In the present paper the authors prove the following useful theorem which allows to apply many results known for convex functions to \(s\)-convex functions. Theorem: Let \(f\) be univalent in \(D\) and \(f(0)=0\). Then \(f\) is \(s\)-convex if and only if the functions \(g_w(z)= f(z)/(1+ \overline w f(z))\) are convex for every \(w\in\overline {f(D)}\). As a consequence of this result they find among others new sharp bounds for \(|f(z)|\), \(|f'(z)|\) and elegant short proofs for old results on \(s\)-convex functions \(f\).