On the best polynomial approximation of analytical functions in the Bergman space $B_2$

In this paper a number of extreme problems related to the best polynomial approximation of analytical in a circle $U:=\{z\in\mathbb{C}:|z|<1\}$ functions belonging to the Bergman's space $B_2$ are being solved. The bilateral inequality is proved, which is a generalization of the result of periodic functions $f\in L_{2}$, by M.Sh.Shabozov–G.A.Yusupov obtained for the class $L_{2}^{(r)}[0,2\pi]$-in which $(r-1)$ the derivative of $f^{(r-1)}$ is absolutely continuous, and the derivative of $r $ is order of $f^{(r)}\ in L_{2}$ in the case of a polynomial approximation of $f\in \mathcal{A}(U)$ belonging to $B_{2}^{(r)}(U)$.
A number of cases are given when the bilateral inequality turns into equality. For some classes of functions belonging to $B_2$, the exact values of the known $n$-diameters are found, and the problem of joint approximation of functions and their intermediate derivatives is solved.