Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 486375, 7 pages
doi:10.1155/2009/486375
Research Article

Superstability of Generalized Multiplicative Functionals

Takeshi Miura,1 Hiroyuki Takagi,2 Makoto Tsukada,3 and Sin-Ei Takahasi1

1Department of Applied Mathematics and Physics, Graduate School of Science and Engineering, Yamagata University, Yonezawa 992-8510, Japan
2Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan
3Department of Information Sciences, Toho University, Funabashi, Chiba 274-8510, Japan

Received 2 March 2009; Accepted 20 May 2009

Academic Editor: Radu Precup

Copyright © 2009 Takeshi Miura et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let X be a set with a binary operation ∘ such that, for each x,y,z∈X, either (x∘y)∘z=(x∘z)∘y, or z∘(x∘y)=x∘(z∘y). We show the superstability of the functional equation g(x∘y)=g(x)g(y). More explicitly, if ε≥0 and f:X→ℂ satisfies |f(x∘y)−f(x)f(y)|≤ε for each x,y∈X, then f(x∘y)=f(x)f(y) for all x,y∈X, or |f(x)|≤(1+1+4ε)/2 for all x∈X. In the latter case, the constant (1+1+4ε)/2 is the best possible.