Let (Formula presented.) be an annulus with boundary circles (Formula presented.) and (Formula presented.) centered at zero; its inner and outer radii are r and R , respectively, (Formula presented.). On the class of functions analytic in the annulus (Formula presented.) with finite (Formula presented.) -norms of the angular limits on the circle (Formula presented.) and of the n th derivatives (of the functions themselves for n0 ) on the circle (Formula presented.), we study interconnected extremal problems for the operator (Formula presented.) that takes the boundary values of a function on (Formula presented.) to its restriction (for m) ) or the restriction of its m th derivative (for mo) to an intermediate circle (Formula presented.) , (Formula presented.). The problem of the best approximation of (Formula presented.) by bounded linear operators from (Formula presented.) to (Formula presented.) is solved. A method for the optimal recovery of the m th derivative on an intermediate circle (Formula presented.) from (Formula presented.)-approximately given values of the function on the boundary circle (Formula presented.) is proposed and its error is found. The Hadamard–Kolmogorov exact inequality, which estimates the uniform norm of the m th derivative on an intermediate circle (Formula presented.) in terms of the (Formula presented.)-norms of the limit boundary values of the function and the n th derivative on the circles (Formula presented.) and (Formula presented.), is derived. © 2023, Pleiades Publishing, Ltd.