On an estimate of integrals for N-sheeted analytic functions

The talk is based on the joint results obtained in co-authorship with A.D. Baranov. They concern the estimation of the integral averages of the derivatives of analytic functions that map a given flat, simply connected domain onto an N-sheeted circle. The problem has a very rich pedigree going back to the classical estimates of integral means Gronwall and Littlewood. Connections of this problem with the inequalities of E.P. Dolzhenko and the Brennan hypothesis.