A univalency criterion (Q5902870)

Let f be meromorphic and locally univalent in the upper half plane U with Schwarzian derivative S(f,z). Suppose that \(| 2(Im z)S(f,z)-c(c- 1)| \leq k| c|\) for all \(z\in U\). It is shown that if \(| c-1| \leq k<1,\) then f is univalent and has a k-quasiconformal extension to the Riemann sphere, and that if \(| c-1| <1\) and \(k=1\), then f is univalent. This was conjectured by \textit{L. V. Ahlfors} in Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, 23- 29 (1974; Zbl 0324.30034).