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

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
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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• http://mi.mathnet.ru/eng/mz1287
• https://doi.org/10.4213/mzm1287
• http://mi.mathnet.ru/eng/mz/v63/i3/p332

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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
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