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Mat. Zametki, 1998, Volume 63, Issue 3, Pages 332–342 (Mi mz1287)  

This article is cited in 1 scientific paper (total in 1 paper)

Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces

V. F. Babenko, V. A. Kofanov, S. A. Pichugov

Dnepropetrovsk State University

Abstract: Suppose that $X$ and $Y$ are real Banach spaces, $U\subset X$ is an open bounded set star-shaped with respect to some point, $n,k\in\mathbb N$, $k<n$, and $M_{n,k}(U,Y)$ is the sharp constant in the Markov type inequality for derivatives of polynomial mappings. It is proved that for any $M\ge M_{n,k}(U,Y)$ there exists a constant $B>0$ such that for any function$f\in C^n(U,Y)$ the following inequality holds:
$$ |\kern -.8pt|\kern -.8pt|f^{(k)}|\kern -.8pt|\kern -.8pt|_U\le M|\kern -.8pt|\kern -.8pt|f|\kern -.8pt|\kern -.8pt|_U+B|\kern -.8pt|\kern -.8pt|f^{(n)}|\kern -.8pt|\kern -.8pt|_U. $$
The constant $M=M_{n-1,k}(U,Y)$ is best possible in the sense that $M_{n-1,k}(U,Y)=\inf M$, where $\inf$ is taken over all $M$ such that for some $B>0$ the estimate holds for all $f\in C^n(U,Y)$.

DOI: https://doi.org/10.4213/mzm1287

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English version:
Mathematical Notes, 1998, 63:3, 293–301

Bibliographic databases:

UDC: 517.5
Received: 17.10.1995
Revised: 29.07.1997

Citation: V. F. Babenko, V. A. Kofanov, S. A. Pichugov, “Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces”, Mat. Zametki, 63:3 (1998), 332–342; Math. Notes, 63:3 (1998), 293–301

Citation in format AMSBIB
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\by V.~F.~Babenko, V.~A.~Kofanov, S.~A.~Pichugov
\paper Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces
\jour Mat. Zametki
\yr 1998
\vol 63
\issue 3
\pages 332--342
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\crossref{https://doi.org/10.4213/mzm1287}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1631920}
\zmath{https://zbmath.org/?q=an:0927.46025}
\transl
\jour Math. Notes
\yr 1998
\vol 63
\issue 3
\pages 293--301
\crossref{https://doi.org/10.1007/BF02317773}
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    This publication is cited in the following articles:
    1. Skorokhodov D.S., “On Inequalities for the Norms of Intermediate Derivatives of Multiply Monotone Functions Defined on a Finite Segment”, Ukr. Math. J., 64:4 (2012), 575–593  crossref  mathscinet  zmath  isi  scopus  scopus
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