On the Landau-Kolmogorov inequality between $\| f' \|_{\infty}$, $\| f \|_{\infty}$ and $\| f''' \|_1$

We solve the Landau-Kolmogorov problem on finding sharp additive inequalities that estimate $\| f' \|_{\infty}$ in terms of $\| f \|_{\infty}$ and $\| f''' \|_1$. Simultaneously we solve related problems of the best approximation of first order differentiation operator $D^1$ by linear bounded ones and the best recovery of operator $D^1$ on elements of a class given with error.

CHEBYSHEV INEQUALITIES WITH LAW-INVARIANT DEVIATION MEASURES

The consistency of law-invariant general deviation measures with concave ordering has been used to generalize the Rao–Blackwell theorem and to develop an approach for reducing minimization of law-invariant deviation measures to minimization of the measures on subsets of undominated random variables with respect to concave ordering. This approach has been applied for constructing the Chebyshev and Kolmogorov inequalities with law-invariant deviation measures—in particular with mean absolute deviation, lower semideviation and conditional value-at-risk deviation. Additionally, an advantage of the Kolmogorov inequality with certain deviation measures has been illustrated in estimating the probability of the exchange rate of two currencies to be within specified bounds.