Exact inequalities between the best polynomial approximations and averaged norms of finite differences in the $B_{2}$ space and widths of some classes of functions

In this paper, exact constants in Jackson–Stechkin type inequalities for characterizing the smoothness of the functions $\Lambda_{m}(f), \ m\in\mathbb{N},$ defined by averaging the norms of finite differences of the $m$-th order of the function $f$ over the argument $z=\rho e^{it}$ analytic in the unit disc belonging $U:=\{z:|z|<1\}$ to the Bergman space $B_{2}$ are found. For the classes of analytic functions in the disk $U$, defined by the characteristics of smoothness $\Lambda_{m}(f)$ and $\Phi$ majorants, satisfying a number of conditions, the exact values of various $n$-widths are calculated.