On Gehring-Pommerenke's estimate for quasicircles
A Jordan curve \(\Gamma\) is called a quasicircle with constant \(M\) if \[ \min \{\text{diam} \Gamma_1, \text{diam} \Gamma_2\} \leq M |w_1 - w_2 |, \] where \(w_1, w_2 \in \Gamma\) and \(\Gamma_1, \Gamma_2\) are the two components of \(\Gamma \backslash \{w_1, w_2\}\). A domain bounded by a quasicircle with constant \(M\) is called a quasidisk with constant \(M\). Assume that \(f(z)\) is meromorphic and locally univalent in the unit disk \(D\), and \(|S_f |_D \leq b < 2\), where \(S_f\) is the Schwarzian derivative of \(f\). In this paper, the author proves that \(f(D)\) is a quasidisk with constant \[ M < 2 \left( 2 + \sqrt {3 + {b \over 6}} \right) \left( 1 - {b \over 2} \right)^{- {1 \over 2}}, \] which improves a result by \textit{F. W. Gehring} and \textit{Ch. Pommerenke} [Comment. Math. Helv. 59, 226-242 (1984; Zbl 0543.30014)], where the estimation for the constant \(M\) is \(M < 8(1 - {b \over 2})^{- {1 \over 2}}\).
scientific article; zbMATH DE number 895927
scientific article