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Functional equations involving the logarithmic mean. (English) Zbl 0885.39008
The authors offer among others a proof that the only solutions $$f$$ of the functional equation $2f(L(x,y))=f(x)+f(y)$ on a proper real interval which are continuous at a point are the constant ones [cf. H. Wundt, Z. Angew. Math. Phys. 5, 172-175 (1954; Zbl 0057.19203)]. Here $L(x,y) = (x-y)/(\ln x-\ln y)$ for $$x\neq y$$ and $$L(x,x)=x$$.

MSC:
 39B22 Functional equations for real functions
Zbl 0057.19203
Full Text:
References:
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