Extension of a generalized Pexider equation
Proc. Amer. Math. Soc. 133 (2005), 3227-3233
June 20, 2005
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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.
Aczél, Lectures on functional equations and their
applications, Mathematics in Science and Engineering, Vol. 19,
Academic Press, New York, 1966. Translated by Scripta Technica, Inc.
Supplemented by the author. Edited by Hansjorg Oser. MR 0208210
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
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
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.
H. Hardy, J.
E. Littlewood, and G.
Pólya, Inequalities, Cambridge, at the University
Press, 1952. 2d ed. MR 0046395
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
Lundberg, On the functional equation
Aequationes Math. 16 (1977), no. 1-2, 21–30. MR 0611541
Radó and John
A. Baker, Pexider’s equation and aggregation of
allocations, Aequationes Math. 32 (1987),
no. 2-3, 227–239. MR 900703
- 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)
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Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
generalized Pexider equation
Received by editor(s):
February 25, 2004
June 20, 2005
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.
M. Gregory Forest
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The copyright for this article reverts to public domain 28 years after publication.