On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment - Skorokhodov, D. S. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
We study the following modification of the Landau-Kolmogorov problem: Let k, r N, 1 ≤ k ≤ r -1 and p, q, s [1, ∞]. Also let MM^m, m N, be the class of nonnegative functions defined on the segment [0,1] whose derivatives of orders 1, 2,... , m are nonnegative almost everywhere on [0,1]. For every δ > 0, find the exact value of the quantity w^k, r_p, q, s(δ; MM^m) := { ||x^(k)||_q : x MM^m, ||x||_p ≤ δ, ||x^(r)||_s ≤ 1} We determine the quantity w^k, r_p, q, s(δ; MM^m) in the case where s = ∞ and m {r, r — 1, r — 2}. In addition, we consider certain generalizations of the above-stated modification of the Landau-Kolmogorov problem.