Estimates for Growth of Derivatives of Analytic Functions Along the Radius

Abstract

We study the radial boundary behavior of functions analytic in a unit disk of the complex plane. 1. The classical Hardy–Littlewood theorem [1] describes the relationship between the smoothness of boundary values of an analytic function on the boundary of a disk of analyticity and the rate of increase in the modulus of its higher-order derivatives. This theorem is an efficient tool for the solution of numerous problems of the theory of functions and the theory of trigonometric series. However, it is often necessary to estimate higher-order derivatives of an analytic function by using solely the information about the modulus of continuity of boundary values of the real part of this function. In the present paper, we consider the following problem: Let a function f be analytic in the disk D := {z ∈ C: |z| < 1} and let the function u := Re f be continuous in D. It is known that, for given n ∈ Z+, the function t �→u(e it ) can be represented in the form