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Title: Mean-value theorem for vector-valued functions (English)
Author: Matkowski, Janusz
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 137
Issue: 4
Year: 2012
Pages: 415-423
Summary lang: English
Category: math
Summary: For a differentiable function ${\bf f}\colon I\rightarrow \mathbb {R}^{k},$ where $I$ is a real interval and $k\in \mathbb {N}$, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $M\colon I^{2}\rightarrow I$ such that$$ {\bf f}(x)-{\bf f}( y) =( x-y) {\bf f}'( M(x,y)) ,\quad x,y\in I, $$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented. (English)
Keyword: Lagrange mean-value theorem
Keyword: mean
Keyword: Darboux property of derivative
Keyword: vector-valued function
MSC: 26A24
MSC: 26E60
idZBL: Zbl 1274.26009
idMR: MR3058273
DOI: 10.21136/MB.2012.142997
Date available: 2012-11-10T20:29:44Z
Last updated: 2020-07-29
Stable URL:
Reference: [1] Berrone, L. R., Moro, J.: Lagrangian means.Aequationes Math. 55 (1998), 217-226. Zbl 0903.39006, MR 1615392, 10.1007/s000100050031
Reference: [2] Matkowski, J.: Mean value property and associated functional equation.Aequationes Math. 58 (1999), 46-59. MR 1714318, 10.1007/s000100050006
Reference: [3] Matkowski, J.: A mean-value theorem and its applications.J. Math. Anal. Appl. 373 (2011), 227-234. Zbl 1206.26032, MR 2684472, 10.1016/j.jmaa.2010.06.057


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