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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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