Kofanov V. Inequalities of Kolmogorov type and extremal problems of analysis

The objects of research are the inequalities of Kolmogorov type. The aim of research is to obtain the new inequalities of Kolmogorov type and solve on this basis extremal problems of analysis. The methods of research are the methods of analysis, approximation theory and theory of functions. Particularly, the methods of comparison of derivatives and rearrangements of functions are used. The new exact inequalities of Kolmogorov type are obtained. General theorems which give connections between inequalities of Kolmogorov type and various problems of analysis are established. On this basis the various problems of analysis are solved. The series of new exact inequalities of Bernstein-Nikolskii type are proved. The new comparison theorems for rearrangements and sigma-rearrangements are obtained. The exact values of the best approximation in the mean of Rimann-Liuville function classes by algebraic polynomials are found. The field of utilization is the extremal problems of analysis and approximation theory.