| 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:
|
http://hdl.handle.net/10338.dmlcz/142997 |
| . |
| 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 |
| . |
