Bernstein-type inequalities for mean $n$-valent functions

Abstract: We derive new integral estimates of the derivatives of mean $n$-valent functions in the unit disk. Our results develop and complement estimates obtained by E.P. Dolzhenko and A.A. Pekarskii, as well as recent inequalities obtained by the authors. As an application, we improve some inverse theorems of rational approximation due to Dolzhenko.