Inner radius of univalence for a strongly starlike domain (Q1396722)

A simply connected proper subdomain \(D\) of the complex plane is called strongly starlike of order \(\alpha\in[0,1]\) with respect to \(w_0\in D\) if the conformal mapping \(\varphi\) from the unit disk \(\mathbb{D}\) to \(D\) with \(\varphi (0)=w_0\) satisfies \(\varphi'(0)\neq 0\) and \[ \left|\arg{z\varphi'(z) \over \varphi (z)-\varphi (0)}\right |\leq {\pi\alpha \over 2},\quad \forall z\in \mathbb{D} \setminus \{0\}. \] Let \(\rho_D(w)|dw|\) denote the Poincaré metric on \(D\). The inner radius of univalence of \(D\) is the largest number \(\sigma= \sigma(D)\) such that the condition \(\sup_{w\in D}{|S_f(w)|\over \rho_D(w)^2} \leq\sigma\) implies univalence of \(f\) on \(D\); here \(f\) is a nonconstant meromorphic function on \(D\) and \(S_f\) is its Schwarzian derivative. The author gives a lower bound for \(\sigma(D)\) in terms of \(\alpha\).