On two point distortion theorems for bounded univalent regular functions (Q1348406)

For a univalent regular (analytic) function mapping the unit disk \(\mathbb D\) into itself, let \[ \Delta_1 f(z)=\frac{1-|z |^2}{1-|f(z)|^2} f^\prime(z),\quad z\in \mathbb D \] and note that \(|\Delta_1 f|\) is invariant under linear (Möbius) transformations. The author proves the inequalities \[ C_1(\rho,\sigma)\leq \left ( |\Delta_1 f(z_1)|^p+|\Delta_1 f(z_2)|^p \right)^{1/p} \leq C_2(\rho,\sigma),\quad z_1,z_2\in \mathbb D, \] where \(\rho,\sigma\) are the hyperbolic (Poincaré) distances between \(z_1\) and \(z_2\), \(f(z_1)\) and \(f(z_2)\) respectively, and \(C_1(\rho,\sigma)\), \(C_2(\rho,\sigma)\) are certain expressions in \(\rho, \sigma\) (explicitly given in the paper). The upper bound is valid for all \(p>0\) and the lower bound is valid (only) when \(p\geq \rho/(\rho-\sigma)\). The author provides a complete description of the equality cases for both inequalities. The proof is based on some results of the same author [Bull. Lond., Math. Soc. 30, 151-158 (1998; Zbl 0921.30016)]. The basic tool is the Jenkins theory of quadratic differentials and his general coefficient theorem.