Two-point boundary distortion estimate for Schwarzian derivative of holomorphic function (Q2795497)

A two-point boundary estimate for the Schwarzian derivative of a function holomorphic in the unit disc is proved. NEWLINENEWLINENEWLINENEWLINE Theorem. Let \(f\) be holomorphic in the unit disc \(\mathbb U\), \(|f(z)|<1\), \(\forall z\in\mathbb U\), and let \(z_1\), \(z_2\) be two different points in \(\mathbb U\). Let \(f\) have the following asymptotics in neighborhoods of \(z_1\) and \(z_2\): NEWLINENEWLINENEWLINE\[NEWLINEf(z)=w_1+a_1(z-z_1)+a_2(z-z_1)^2+a_3(z-z_1)^3+o((z-z_1)^3),\quad z\rightarrow z_1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEf(z)=w_2+b_1(z-z_2)+b_2(z-z_2)^2+b_3(z-z_2)^3+o((z-z_2)^3),\quad z\rightarrow z_2,NEWLINE\]NEWLINE where \(w_1\neq w_2\), \(|w_1|=|w_2|=1\), \(a_1b_1\neq 0\), and the ``smallness'' \(o((z-z_k)^3)\), \(z\rightarrow z_k\), is valid with respect to any Stolz angle in \(\mathbb U\) with the vertex \(z_k\). NEWLINENEWLINENEWLINENEWLINE Assume that NEWLINE\[NEWLINE2\operatorname{Re}\frac{z_1a_2}{a_1}=|a_1|-1,\quad 2\operatorname{Re}\frac{z_2 b_2}{b_1}=|b_1|-1,\eqno{(1)}NEWLINE\]NEWLINE and that the image \(f(\mathbb U\setminus\gamma(z_1,z_2))\) does not contain any point of an open arc of the circle with end-points \(w_1, w_2\), where \(\gamma(z_1,z_2)=\{z:|z-z_1|=|z-z_2|\}\). NEWLINENEWLINENEWLINENEWLINE Then NEWLINE\[NEWLINE\operatorname{Re}\{S_f(z_1)+S_f(z_2)\}\leq 12\left[\frac{1}{|z_1-z_2|^2}-\frac{|a_1b_1|}{|w_1-w}\right]. \eqno{(2)}NEWLINE\]NEWLINE Here NEWLINE\[NEWLINES_f(z_1)=6 \left(\frac{a_3}{a_1}-\frac{a_2^2}{a_1^2}\right),\quad S_f(z_2)=6\left(\frac{b_3}{b_1} - \frac{b_2^2}{b_1^2}\right). NEWLINE\]NEWLINENEWLINENEWLINEIn (2) equality takes place, in particular, for any fractional-linear automorphism of the unit disc.