On the best approximation of analytic in a disk functions in the weighted Bergman space $\mathscr{B}_{2,\mu}$

We obtain sharp inequalities between the best approximations of analytic in the unit disk functions by algebraic complex polynomials and the moduli of continuity of higher-order derivatives in the Bergman weighted space $\mathscr{B}_{2,\mu}$. Based on these inequalities, the exact values of some known $n$-widths of classes of analytic in the unit disk functions are calculated.