  In this paper we establish some new Kolmogorov type inequalities for the
Marchaud and Hadamard fractional derivatives of functions defined on a real
axis or semi-axis. Simultaneously we solve two related problems: the Stechkin
problem on the best approximation of unbounded operators by bounded ones on a
given class of elements and the problem of optimal recovery of operator on
elements from some class given with prescribed error. Keywords: inequalities
for derivatives, fractional derivatives, approx- imation of unbounded operators
by bounded ones, optimal recovery of operators, ideal lattice.
