Bojanov-Naidenov problem for positive (negative) parts of differentiable functions on the real domain

V.V. Kameneva; V.A. Kofanov

doi:10.15421/241804

Bojanov-Naidenov problem for positive (negative) parts of differentiable functions on the real domain

V.V. Kameneva (Oles Honchar Dnipro National University)
V.A. Kofanov (Oles Honchar Dnipro National University), https://orcid.org/0000-0003-0392-2257

Abstract

We solve the extremal problem $$$\| x^{(k)}_{\pm} \|_{L_p[a,b]} \rightarrow \sup$$$, $$$k = 0, 1, ..., r-1$$$, over the set of pairs $$$(x, I)$$$ of functions $$$x\in W^r_{\infty} (\mathbb{R})$$$ and intervals $$$I = [a,b]$$$ with restrictions on the local norm of function $$$x$$$ and the measure of support $$$\mu \{ \mathrm{supp}_{[a,b]} x^{(k)}_{\pm} \}$$$.

Keywords

Bojanov-Naidenov problem; positive (negative) part of function; rearrangement; comparison theorem

MSC 2020

26D10; 46E30

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