On coefficient multipliers for area Privalov classes

Abstract

The problem of describing the Taylor coefficients of functions analytic in a disk was first resolved for the Nevanlinna class by an outstanding Soviet mathematician S.N. Mergelyan in the beginning of 20th century. Later, the studies devoted to obtaining similar estimates in various classes of analytic functions were made by known Russian and foreign specialists in the complex analysis: G. Hardy, J. Littlewood, A.A. Friedman, N. Yanagihara, M. Stoll, S.V. Shvedenko and others. In the paper we introduce a area Privalov class $\tilde{\Pi}_q$, $(q>0)$, being a generalization of a known area Nevanlinna class. In the first part of the paper we obtain a sharp estimate for the growth of an arbitrary function in the area Privalov class, we describe the Taylor coefficients for this function. In the second part of the work, on the base of the obtained estimates we describe completely the coefficient multipliers from area Privalov classes into the Hardy classes. In a simplified form this problem can be formulated as follows: by what factors the Taylor coefficients of a function in a given class $\tilde{\Pi}_q$, $q>0$, should be multiplied in order to get the Taylor coefficients of a function in a Hardy class.