Approximation of the differentiation operator on the class of functions analytic in an annulus

In the class of functions analytic in the annulus $C_r:=\left\{z\in\mathbb{C}\, :\, r<|z|<1\right\}$ with bounded $L^p$-norms on the unit circle, we study the problem of the best approximation of the operator taking the nontangential limit boundary values of a function on the circle $\Gamma_r$ of radius $r$ to values of the derivative of the function on the circle $\Gamma_\rho$ of radius $\rho,\, r<\rho<1,$ by bounded linear operators from $L^p(\Gamma_r)$ to $L^p(\Gamma_ \rho)$ with norms not exceeding a number $N$. A solution of the problem has been obtained in the case when $N$ belongs to the union of a sequence of intervals. The related problem of optimal recovery of the derivative of a function from boundary values of the function on $\Gamma_\rho$ given with an error has been solved.