On distortion under quasiconformal mapping (Q1880987)

Let \(S\) be the Riemann sphere with punctures at \(0\) and \(1\). For \(K\geq 1\), let \(Q_K\) denote the class of univalent functions on \(S\) which are \(K\)-quasiconformal which satisfy \(f(S)=S\), \(f(0)=0\) and \(f(1)=1\) and let \(F(f)\) be a continuous functional (or system of functionals) over \(Q_K\). The author calls the ``extremal problem'' for \(F(f)\) the problem of finding \(\max_{f\in Q_{K}}\pm F(f)\). This problem has already been considered by Lehto, Virtanen, Belinski, Lavret'ev and others. The author in this paper studies this extremal problem associated to the system of functionals \((| f(z_1)|, | f(z_2)|)\), defined on the class of \(K\)-quasiconformal homeomorphisms of the Riemann sphere with the normalization \(f(0)=0\), \(f(1)=1\) and \(f(\infty)=\infty\). He describes the extremal functionals in terms of the coefficients of the corresponding Beltrami equation. As a corollary, he obtains some sharp estimates for the conformal functionals \(| f(z_2)| \pm | f(z_1)|\) and \(| f(z_2)\pm f(z_1)|\). The main tool is the extremal partition of the surface by doubly connected domains. The paper ends with a list of interesting open problems.