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In this paper, we derive a generalized multiplicative Hardy-Littlewood-Polyatype inequality, as well as several related additive inequalities, forfunctions of operators in Hilbert spaces. In addition, we find the modulus ofcontinuity of a function of an operator on a class of elements defined with thehelp of another function of the operator. We then apply the results to solvethe following problems: (i) the problem of approximating a function of anunbounded self-adjoint operator by bounded operators, (ii) the problem of bestapproximation of a certain class of elements from a Hilbert space by anotherclass, and (iii) the problem of optimal recovery of an operator on a class ofelements given with an error.

Inequalities of Hardy-Littlewood-Polya type for functions of operators and their applications