On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class.

Abstract

We study quasi-monotonic functions of the Zygmund-Bary-Stechkin class $\Phi$ with the main emphasis on properties of the index numbers of functions in this class. A special attention is paid to functions whose lower and upper index numbers do not coincide with each other (non-equilibrated functions). It is proved that the bounds for functions in $\Phi$ known in terms of these indices, are exact in a certain sense. We also single out some special family of none-equilibrated functions in $\Phi$ which oscillate in a certain way between two power functions. Given two numbers $0< \alpha\leq \beta <1$, we explicitly construct examples of functions in $\Phi$ for which $\alpha$ and $\beta$ serve as the index numbers. The investigation of properties of non-equilibrated functions in $\Phi$ was evoked by applications of these properties in problems of the normal solvability of some singular integral operators in the spaces with prescribed modulus of continuity.