On additive inequalities for intermediate derivatives of functions given on a finite interval

Abstract

We present a general scheme for deducing additive inequalities of Landau-Hadamard type. As a consequence, we prove several new inequalities for the norms of intermediate derivatives of functions given on a finite interval with an exact constant with the norm of a function.

The Landau-Kolmogorov Problem on a Finite Interval in the Taikov Case

“…For modulus Ω t and set Γ t similar relations hold true 4) can be found in Bur_80 [18]. We also refer the reader to the papers Sha_93, BabKofPic_97 [24,19] and books KwoZet_92,BabKorKofPic_03 [22,23] for the overview of results on inequalities (…”

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

“…In [27,28], a general scheme of finding additive Kolmogorovtype inequalities (aimed mainly at the case of functions defined on a segment) was given. The analysis of the proof of inequality (3.3) allows us to propose the following modification of this scheme (which may turn out to be useful for finding other Kolmogorov-type inequalities for functions of many variables):…”

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

“…Partial solutions of Problems 1 and 2 can be found in the papers [11,18,26,4,15,10,27]. For the overview of other results in this and closely related directions we refer the reader to books [20,5] and surveys [2,26].…”

“…Problem 4. For every δ > 0, find the error of the best recovery of operator A with the help of the set of operators (called methods of recovery) R on the elements of the class W given with the error δ: 4) and the best methods of recovery S * ∈ R (if any exists) delivering the infimum in the right hand part of (1.4).…”