Jackson–Stechkin type inequalities between the best joint polynomials approximation and a smoothness characteristic in Bergman space

We consider the extremal problem of finding the exact constants between the best joint polynomial approximations of analytic functions and their intermediate derivatives in the Bergman space. Let $U:=\{z:|z|<1\}$ be the unit disc on the complex plane, $B_{2}:=B_{2}(U)$ the Bergman space of functions $f$ analytic in the disc with finite $L_2$ norm; $B_{2}^{(r)}:=B_{2}^{(r)}(U)$ ($r\in\mathbb{Z}_{+}$, $ B_{2}^{(0)}:=B_{2}$) is a class of functions $f\in B_{2}$, for which $f^{(r)}\in B_{2}$. In this paper, exact constants in Jackson–Stechkin type inequalities for $\Lambda_{m}(f)$, $m\in\mathbb{N}$, the smoothness characteristic determined by averaging the norms of finite differences of the $m$-th order of the highest derivative of a function $f$ belonging to the Bergman space $B_{2}$ are found. Also we solve the extremal problem of the best joint polynomial approximation of the class $W_{2,m}^{(r)}(\Phi):=W_{2}^{(r)}(\Lambda_{m},\Phi)$ ($m\in\mathbb{N}$, $r\in\mathbb{Z}_{+}$) of functions from $B_{2}^{(r)}$, $r\in \mathbb{Z}_{+}$, for which the value of the smoothness characteristic $\Lambda_{m}(f)$ is bounded from above by the majorant $\Phi$ and the class $W_{p,m}^{(r)}(\varphi,h):=W_{p}^{(r)}(\Lambda_{m},\varphi,h)$ ($m\in\mathbb{N}$, $r\in\mathbb{Z}_{+},$ $h\in[0,2\pi],$ $0<p\le\infty,$ $\varphi$ is a weighted function on $[0,h]$) from $B_{2}$, for which the value of the smoothness characteristics of the $\Lambda_{m}(f)$ averaged with a given weight, is bounded from above by one. It should be noted that the results presented in the article are generalizations of the recently published results of the second author [10] for the joint approximation of periodic functions by trigonometric polynomials to the case of joint approximation of functions analytic in the unit circle by complex algebraic polynomials in the Bergman space.