| Title:
|
Some logarithmic functional equations (English) |
| Author:
|
Laohakosol, Vichian |
| Author:
|
Pimsert, Watcharapon |
| Author:
|
Hengkrawit, Charinthip |
| Author:
|
Ebanks, Bruce |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
173-181 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The functional equation $f(y-x) - g(xy) = h\left(1/x-1/y\right)$ is solved for general solution. The result is then applied to show that the three functional equations $f(xy)=f(x)+f(y)$, $f(y-x)-f(xy)=f(1/x-1/y)$ and $f(y-x)-f(x)-f(y)=f(1/x-1/y)$ are equivalent. Finally, twice differentiable solution functions of the functional equation $f(y-x) - g_1(x)-g_2(y) = h\left(1/x-1/y\right)$ are determined. (English) |
| Keyword:
|
logarithmic functional equation |
| Keyword:
|
Pexider equations |
| MSC:
|
39B20 |
| idMR:
|
MR2995870 |
| DOI:
|
10.5817/AM2012-3-173 |
| . |
| Date available:
|
2012-10-03T14:46:21Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142987 |
| . |
| Reference:
|
[1] Chung, J.–Y.: A remark on a logarithmic functional equation.J. Math. Anal. Appl. 336 (2007), 745–748. Zbl 1130.39018, MR 2348539, 10.1016/j.jmaa.2007.02.072 |
| Reference:
|
[2] Ebanks, B.: On Heuvers’ logarithmic functional equation.Result. Math. 42 (2002), 37–41. Zbl 1044.39018, MR 1934223, 10.1007/BF03323552 |
| Reference:
|
[3] Heuvers, K. J.: Another logarithmic functional equation.Aequationes Math. 58 (1999), 260–264. MR 1715396, 10.1007/s000100050112 |
| Reference:
|
[4] Heuvers, K. J., Kannappan, P.: A third logarithmic functional equation and Pexider generalizations.Aequationes Math. 70 (2005), 117–121. Zbl 1079.39019, MR 2167989, 10.1007/s00010-005-2792-8 |
| Reference:
|
[5] Kannappan, P.: Functional Equations and Inequalities with Applications.Springer, Dordrecht, 2009. Zbl 1178.39032, MR 2524097 |
| Reference:
|
[6] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities.second ed., Birkhäuser, Basel, 2009. Zbl 1221.39041, MR 2467621 |
| Reference:
|
[7] Rätz, J.: On the theory of functional equation $f(xy) = f(x)+f(y)$.Elem. Math. 21 (1966), 10–13. |
| . |
