Strongly hyperbolically convex functions (Q2642197)

Let \(C(w_1,w_2,w_3)\) denote the circle in \(\widehat{\mathbb C}\) through \(w_1,w_2,w_3\), and let \(\widehat{w_1w_2}\) denote one of the two arcs between \(w_1,w_2\) belonging to \(C(w_1,w_2,w_3)\). The authors prove that a domain \(\Omega\) in the Riemann sphere with no antipodal points is spherically convex if and only if for any \(w_1,w_2,w_3\in \Omega\) with \(w_1\neq w_2\), the arc \(\widehat{w_1w_2}\) of the circle \(C(w_1,w_2,-1/\overline{w_3})\) which does not contain \(-1/\overline{w_3}\) lies in \(\Omega\). Based on this characterization they call a domain \(G\) in the unit disk \(\mathbb{D}\) strongly hyperbolically convex if for any \(w_1,w_2,w_3\in G\) with \(w_1\neq w_2\), the arc \(\widehat{w_1w_2}\) in \(\mathbb{D}\) of the circle \(C(w_1,w_2,-1/\overline{w_3})\) is also contained in \(G\). The authors obtain number of results on conformal maps \(f\) onto strongly hyperbolically convex domains, including bounds on the hyperbolic derivative \(| f'| /(1-| f| ^2)\), and bounds on the pre-Schwarzian and Schwarzian derivative.