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Superstability of generalized multiplicative functionals. (English) Zbl 1176.39023
Summary: Let $$X$$ be a set with a binary operation $$\circ$$ such that, for each $$x,y,z\in X$$, either $$(x\circ y)\circ z=(x\circ z)\circ y$$, or $$z\circ(x\circ y)=x\circ(z\circ y)$$. We show the superstability of the functional equation $$g(x\circ y)=g(x)g(y)$$. More explicitly, if $$\varepsilon\geq 0$$ and $$f:X\to\mathbb C$$ satisfies $$|f(x\circ y)-f(x)f(y)|\leq\varepsilon$$ for each $$x,y\in X$$, then $$f(x\circ y)=f(x)f(y)$$ for all $$x,y\in X$$, or $$|f(x)|\leq (1+\sqrt{1+4\varepsilon})/2$$ for all $$x\in X$$. In the latter case, the constant $$(1+\sqrt{1+4\varepsilon})/2$$ is the best possible.
MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
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