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Bounded Solutions of a Functional Inequality

Published online by Cambridge University Press:  20 November 2018

Michael Albert  and
John A. Baker
Michael Albert
Affiliation:
Department of Pure Mathematics,University of Waterloo, Waterloo, Ontario Canada, N2L 3G1
John A. Baker
Affiliation:
Department of Pure Mathematics,University of Waterloo, Waterloo, Ontario Canada, N2L 3G1
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Abstract

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It is known that if f is a real valued function on a rational vector space V, δ > 0,

1

and if f is unbounded then f(x + y) = f(x)f(y) for all x, y ∊ V. In response to a problem of E. Lukacs, in this paper we study the bounded solutions of (1). For example, it is shown that if f is a bounded solution of (1) then - δ ≤ f(x) ≤ (1 + (1 + 4δ)1/2)/2 for all x ∊ V and these bounds are optimal.

Keywords

39C0526D9939B20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 4 , 01 December 1982 , pp. 491 - 495
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Baker, John, Lawrence, J. and Zorzitto, F., The Stability of the Equation f(x + y) = f(x)f(y), Proc. Amer. Math. Soc. 74 (1979), 242-246.Google Scholar
2. Baker, John A., The Stability of the Cosine Equation, Proc. Amer. Math. Soc. 80 (1980), 411-416.Google Scholar