Estimates for integrals of derivatives of $n$-valent functions and geometric properties of domains

Abstract

A number of questions concerning the behaviour of double integrals of the moduli of the derivatives of bounded $n$-valent functions and, in particular, of rational functions of fixed degree $n$ are considered. For domains with rectifiable boundaries the sharp order of growth of such integral means is found in its dependence on $n$. Upper bounds for domains with fractal boundaries are obtained, which depend on the Minkowski dimension of the boundary of the domain. In certain cases these bounds are shown to be close to sharp ones. Lower bounds in terms of the integral means spectra of conformal mappings are also found. These inequalities refine Dolzhenko's classical results (1966) and some recent results due to the authors. Bibliography: 32 titles.