Some inequalities between the best simultaneous approximation and the modulus of continuity in a weighted Bergman space

Some inequalities between the best simultaneous approximation of functions and their intermediate derivatives, and the modulus of continuity in a weighted Bergman space are obtained. When the weight function is $\gamma(\rho)=\rho^\alpha,\ \alpha>0$, some sharp inequalities between the best simultaneous approximation and an $m$th order modulus of continuity averaged with the given weight are proved. For a specific class of functions, the upper bound of the best simultaneous approximation in the space $B_{2,\gamma_{1}},$ $\gamma_{1}(\rho)=\rho^{\alpha},\ \alpha>0$, is found. Exact values of several $n$-widths are calculated for the classes of functions $W_{p}^{(r)}(\omega_{m},q)$.