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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:
[1] : Lectures on functional equations and their applications. Academic Press, New York and London 1966. · Zbl 0139.09301
[2] : Functional equations in several variables. Cambridge University Press, Cambridge–New York 1989.
[3] ; ; : Means and their inequalites. D. Reidel Publishing Co., Dordrecht–Boston-Lancaster–Takyo 1988.
[4] Hosszú, Z. angew. Math. Phys. 6 pp 143– (1955)
[5] Kahlig, Aequationes Math. 41 pp 304– (1991)
[6] : Functional equations in a single variable. Monografie Mat. 46, PWN, Warszawa 1968.
[7] Kuczma, Rozprawy Math. (Dissertationes Math.) 34 (1963)
[8] ; ; : Iterative functional equations. Cambridge University Press, Cambridge–New York 1990.
[9] Lundberg, Arkiv för Mat. 5 pp 193– (1963)
[10] Matkowski, Aequationes Math. 40 pp 168– (1990)
[11] : Mechanika a termomechanika pro elektroenergetika. SNTL/ALFA, Praha 1987.
[12] : Dynamik der Atmosphäre. BI, Mannheim–Wein–Zürich 1986.
[13] Stiefel, Hosszú. Z. angew. Math. Phys. 6 pp 144– (1955)
[14] Wundt, Z. angew. Math. Phys. 5 pp 172– (1954)
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