Best approximation of the differentiation operator in spaces of functions analytic in a strip

In spaces of functions analytic in a strip, we obtain sharp estimates of the $L^p$-norms, $1\leqslant p\leqslant\infty$, of the derivative on an intermediate line in terms of the $L^p$-norms of the limit values of the function on the boundary lines. We also consider the Stechkin problem of best approximation of the differentiation operator by bounded operators.