On rate of decreasing of extremal function in Carleman class

Abstract

We study the issues related with Levinson-Sjöberg-Wolf type theorems in the complex analysis and, in particular, we discuss a famous question posed in 70s by E.M. Dyn'kin on an effective bound for majorant of the growth of an analytic function in the vicinity of the set of singular points and another close problem on the rate of decaying of an extremal function in a non-quasianalytic Carleman class in the vicinity of the point, at which all the derivatives of the functions from this class vanish. Exact asymptotic estimates for the best majorant for the growth in the vicinity of the singularities were found by V. Matsaev and M. Sodin in 2002. Some bounds, both from above and below, for an extremal function in the Carleman class were obtained by A.M. Gaisin in 2018 but they turned out to be not very close to exact values of this function. In the present paper we obtain sharp two-sided estimates for the extremal function.