Extension of a generalized Pexider equation
Abstract: The equations and , called Pexider equations, have been completely solved on If they are assumed to hold only on an open region, they can be extended to (the second when is nowhere 0) and solved that way. In this paper their common generalization is extended from an open region to and then completely solved if is not constant on any proper interval. This equation has further interesting particular cases, such as and that arose in characterization of geometric and power means and in a problem of equivalence of certain utility representations, respectively, where the equations may hold only on an open region in Thus these problems are solved too.
- 1. J. Aczél, Lectures on functional equations and their applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York-London, 1966. Translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser. MR 0208210
- 2. J. Aczél, A short course on functional equations, Theory and Decision Library. Series B: Mathematical and Statistical Methods, D. Reidel Publishing Co., Dordrecht, 1987. Based upon recent applications to the social and behavioral sciences. MR 875412
- 3. J. Aczél and J. K. Chung, Integrable solutions of functional equations of a general type, Studia Sci. Math. Hungar. 17 (1982), no. 1-4, 51–67. MR 761524
- 4. A. Gilányi, C.T. Ng and J. Aczél, On a functional equation arising from comparison of utility representations, J. Math. Anal. Appl. 304 (2005), 572-583.
- 5. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- 6. A. Járai, A remark to a paper of J. Aczél and J. K. Chung: “Integrable solutions of functional equations of a general type” [Studia Sci. Math. Hungar. 17 (1982), no. 1-4, 51–67; MR0761524 (85i:39008)], Studia Sci. Math. Hungar. 19 (1984), no. 2-4, 273–274. MR 874494
- 7. A. Lundberg, On the functional equation 𝑓(𝜆(𝑥)+𝑔(𝑦))=𝜇(𝑥)+ℎ(𝑥+𝑦), Aequationes Math. 16 (1977), no. 1-2, 21–30. MR 0611541
- 8. F. Radó and John A. Baker, Pexider’s equation and aggregation of allocations, Aequationes Math. 32 (1987), no. 2-3, 227–239. MR 900703, 10.1007/BF02311311
- J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York/London, 1966. MR 0208210 (34:8020)
- J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Reidel/Kluwer, Dordrecht/Boston, 1987. MR 0875412 (88d:39013)
- J. Aczél and J.K. Chung, Integrable solutions of functional equations of a general type, Stud. Sci. Math. Hungar. 17 (1982), 51-67. MR 0761524 (85i:39008)
- A. Gilányi, C.T. Ng and J. Aczél, On a functional equation arising from comparison of utility representations, J. Math. Anal. Appl. 304 (2005), 572-583.
- G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, Cambridge/London, 1951. MR 0046395 (13:727e)
- A. Járai, A remark to a paper of J. Aczél and J.K. Chung: ``Integrable solutions of functional equations of a general type'', Stud. Sci. Math. Hungar. 19 (1984), 273-274. MR 0874494 (87m:39006)
- A. Lundberg, On the functional equation , Aequationes Math. 16 (1977), 21-30. MR 0611541 (58:29529)
- F. Radó and J.A. Baker, Pexider's equation and aggregation of allocations, Aequationes Math. 32 (1987), 227-239. MR 0900703 (89i:39014)
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 39B22
Retrieve articles in all journals with MSC (2000): 39B22
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Keywords: Functional equations, extensions, generalized Pexider equation
Received by editor(s): February 25, 2004
Published electronically: June 20, 2005
Additional Notes: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada grant OGP 0002972. The author is grateful for an observation by Fulvia Skof.
Communicated by: M. Gregory Forest
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.