  In this paper, we derive a generalized multiplicative Hardy-Littlewood-Polya
type inequality, as well as several related additive inequalities, for
functions of operators in Hilbert spaces. In addition, we find the modulus of
continuity of a function of an operator on a class of elements defined with the
help of another function of the operator. We then apply the results to solve
the following problems: (i) the problem of approximating a function of an
unbounded self-adjoint operator by bounded operators, (ii) the problem of best
approximation of a certain class of elements from a Hilbert space by another
class, and (iii) the problem of optimal recovery of an operator on a class of
elements given with an error.
