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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

Citations:

Zbl 0057.19203
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References:

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