Meremelia I. Best approximations and extremal problems of classes of holomorphic functions

In the thesis there was determined the asymptotic equality for the upper bounds of deviations of generalized Zygmund sums on some functional classes that are convolution of unit ball of the Hardy space in case when a sequence which defined classes are the moment. And also were given necessary and sufficient conditions under which the generalized Zygmund sums approximate the class H^{psi phi}_p with minimal possible error. The behavior on the interval [0,1] the majorant of the n-remainders of Taylor's series for bounded holomorphic functions in the unit disk is studied. The exact constants in the inequalities for the pairs of Taylor coefficients on some classes of bounded holomorphic functions in a polydisk are determined. The solution of Pompeiu-Landau-Szasz's problem for partial sums of Taylor series of bounded holomorphic functions in the bidisk is finded. For the Hardy space of holomorphic functions in the polydisk the exact inequalities of mixed derivatives is obtained.