Stechkin's problem for differential operators and functionals of first and second orders

In this paper, we present the solution to Stechkin's problem for differential operators and functionals of first and second orders on the class of functions that are defined on a finite interval and have bounded third derivative. Two related problems are discussed: (1) finding sharp constants in the Landau-Kolmogorov inequalities, and (2) optimal recovery of an operator with the help of a set of linear operators (or arbitrary single-valued mappings) on elements of some set that are given with an error.