Schwarzian norms and two-point distortion (Q411888)

For a function \(f\) analytic and locally univalent in the unit disk \(\mathbb{D}\), its Schwarzian derivative in \(\mathbb{D}\) is given by \({\mathcal S}f:= (f^n/f')'-{1\over 2}(f^n/f')^2\) and its Schwarzian norm is defined to be \(\|{\mathcal S}f\|:= \sup\{(1-|z|^2)^2|{\mathcal S}f|: z\in\mathbb{D}\}\). The authors extend earlier results of \textit{B. Schwarz} [Trans. Am. Math. Soc. 80, 159--186 (1955; Zbl 0067.31703)] and \textit{M. Chuaqui} and \textit{C. Pommerenke} [Pac. J. Math. 188, No. 1, 83--94 (1999; Zbl 0931.30016)] by showing that if \(\|{\mathcal S}f\|\leq 2(1+\delta^2)\), \(\delta> 0\), then \(f\) satisfies a pair of two-point distortion conditions, one a lower bound and one an upper bound for the distortion NEWLINE\[NEWLINE\Delta_f(\alpha, \beta):= |f(\alpha)- f(\beta)|/\{(1- |\alpha|^2)|f'(\alpha)|(1- |\beta|^2)| f'(\beta)|\}^{1/2};NEWLINE\]NEWLINE and, conversely, that each of these conditions implies that \(\|{\mathcal S}f\|\leq 2(1+\delta^2)\), \(\delta>0\).NEWLINENEWLINE Finally, they obtain analogues of the lower bound for curves in \(\mathbb{R}^n\) and for canonical lifts of harmonic mappings to minimal surfaces.