| Title:
|
Some mean value theorems as consequences of the Darboux property (English) |
| Author:
|
Marinescu, Dan Ştefan |
| Author:
|
Monea, Mihai |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
142 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
211-224 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals. (English) |
| Keyword:
|
Darboux function |
| Keyword:
|
mean value theorem |
| Keyword:
|
continuous function |
| Keyword:
|
integrable function |
| Keyword:
|
differentiable function |
| Keyword:
|
arithmetic mean |
| Keyword:
|
geometric mean |
| Keyword:
|
harmonic mean |
| MSC:
|
26A15 |
| MSC:
|
26A24 |
| MSC:
|
26A42 |
| idZBL:
|
Zbl 06738581 |
| idMR:
|
MR3660177 |
| DOI:
|
10.21136/MB.2016.0032-15 |
| . |
| Date available:
|
2017-05-23T10:01:52Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146754 |
| . |
| Reference:
|
[1] Chen, Z.: A higher mean value theorem.Amer. Math. Monthly 110 (2003), 544-545. 10.2307/3647923 |
| Reference:
|
[2] Garcia, T. M., Suarez, P.: Solution of problem 1867.Mathematics Magazine 85 (2012), page 153. Zbl 1246.97017, MR 3324702 |
| Reference:
|
[3] Herman, E., Lampakis, E., Witkowski, A.: Solution of problem 956.Coll. Math. J. 43 (2012), 338-339. |
| Reference:
|
[4] Jarník, V.: Über die Differenzierbarkeit stetiger Funktionen.Fundam. Math. 21 (1933), 48-58. Zbl 0007.40102, 10.4064/fm-21-1-48-58 |
| Reference:
|
[5] Kowalewski, G.: Interpolation und genäherte Quadratur. Eine Ergänzung zu den Lehr- büchern der Differential- und Integralrechnung.B. G. Teubner, Leipzig und Berlin (1932). Zbl 0004.05605 |
| Reference:
|
[6] Marinescu, D. Ş.: În legătură cu o problemă de concurs.Recreaţii Matematice 1 (2004), 20-22 (in Romanian). |
| Reference:
|
[7] Marinescu, D. Ş.: Problem 26546.Gazeta Matematică, seria B 12 (2001). |
| Reference:
|
[8] Marinescu, D. Ş., Monea, M., Stroe, M.: Teorema lui Jarnik şi unele consecinţe.Revista de Matematică a Elevilor din Timişoara 4 (2010), 3-8 (in Romanian). |
| Reference:
|
[9] Orno, P.: Problem 1053.Mathematics Magazine 51 (1978), page 245. MR 1572276, 10.2307/2689475 |
| Reference:
|
[10] Pangsriiam, P.: Problem 11753.American Mathematical Monthly 121 (2014), page 84. |
| Reference:
|
[11] Plaza, A., Rodriguez, C.: Problem 1867.Mathematics Magazine 84 (2011), page 150. |
| Reference:
|
[12] Precupanu, T.: Problem 5.119.Olimpiadele Naţionale de Matematică 1954-2003 (D. Bă- tineţu, I. Tomescu, eds.). Ed. Enciclopedică, Bucureşti (2004), page 146 (in Romanian). |
| Reference:
|
[13] C. F. Rocca, Jr.: A question of integral.Mat: 450 Senior Seminar (2012). |
| Reference:
|
[14] Sahoo, P. K., Riedel, T.: Mean Value Theorems and Functional Equations.World Scientific, Singapore (1998). Zbl 0980.39015, MR 1692936 |
| Reference:
|
[15] Thong, Duong Viet: Problem 956.Coll. Math. J. 42 (2011), page 329. |
| . |
