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Under some natural assumptions on real functions f and g defined on a real interval I, we show that a two variable function Mf,g : I2 → I defined by $$M_{f,g}(x,y)=(f+g)^{-1}(f(x)+g(y))$$ is a generalization of the quasi-arithmetic mean. Necessary and sufficient conditions for: symmetry, quasi-arithmeticity, weighted quasi-arithmeticity, homogeneity of Mf,g, as well as equality of two...
We deal with the problem of determining general solutions $${f\colon\mathbb{R}\to\mathbb{R}}$$ of the following composite functional equation introduced by Fechner: $$ f(f(x)-f(y))=f(x+y)+f(x-y)-f(x)-f(y). $$ Our result gives a partial answer to this problem under some assumptions upon $${f(\mathbb{R})}$$ . We are applying a theorem of Simon and Volkmann concerning a certain characterization...
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