Inequalities of Kolmogorov type for analytic functions of one and two complex variables and their applications to approximation theory

Abstract

UDC 517.5 Exact inequalities of the Kolmogorov type are obtained in Hardy Banach spaces for functions of one complex variable analytic in the unit disk and functions of two complex variables analytic in the unit bidisk. We also present applications of these inequalities to problems of the theory of approximation of analytic functions of one and two complex variables. Since the beginning of the 20th century, there has been much interest among mathematicians, beginning with Landau, Hadamard, Hardy, Littlewood, and Kolmogorov, in finding exact inequalities for norms of intermediate derivatives of functions in terms of the norm of the function itself and the norm of its higher-order derivative. The modern advances in the study of this problem are associated with works by Arestov, Stechkin, Taikov, Gabushin, Tikhomirov, Kuptsov, Konovalov, Korneichuk, Babenko, Magaril-Il’yaev, Ligun, Pichugov, Kofanov, and many others (see, e.g., [1] and the bibliography therein). In our opinion, of no less interest is the solution of analogous problems in the case of analytic functions of complex variables, where not as many unimprovable results are available as in the case of real variables (see e.g. [2‐6]). The present paper continues these investigations in the complex plane.