Quasiconformal extension and univalency criteria (Q1199262)

Extending earlier work of Ahlfors, Anderson and Hinkkanen, Harmelin, the author proves the following injectivity criterion for analytic functions defined in the upper half-plane \(U\). Theorem. Let \(f(z)\) and \(a(z)\) be analytic functions in \(U\) with \(f'(z)\neq 0\) for all \(z\in U\) and let \(c\in\mathbb{C}\backslash\{0\}\). Suppose that \[ |(\overline z- z)a(z)+cf'(z)e^{-\int a(z)dz}-1|\leq k \] for all \(z\in U\). If \(k<1\), then \(f(z)\) is univalent in \(U\) and has a \(k\)-quasiconformal extension to \(\mathbb{C}\). If \(k=1\), then \(f(z)\) is univalent in \(U\).