On distortion under quasiconformal mapping

In the paper we study the range of the system of functionals (\f(Z(1))\, \f(z(2))\) over the class of K-quasiconformal homeomorphisms of the Riemann sphere with standard three point normalization f(0) = 0, f(1) = 1, f(infinity) = infinity, and for different real values of z(1) and z(2). Extremal functions are given in terms of the complex dilatation dependent only on z(1), z(2). As a corollary, we derive some sharp estimates for the functional \f(z(2))\ +/- \f(z(1))\ and \f(z(2)) +/- f(z(1))\. The main tool of the proofs is the extremal partition of a Riemann surface by doubly connected domains.