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On two functional equations related to a result of Grégoire de Saint-Vincent. (English) Zbl 0836.39005

Let \(f\) be a continuous function from \((- \infty, + \infty)\) into \((0, + \infty)\). The following conditions are equivalent: \[ f \left( {x + 2y \over 3} \right) f \left( {2x + y \over 3} \right) = f(x) f(y) \] for all \(x,y\) in \(\mathbb{R}\), \[ f(x)f \left( {x + 2y \over 3} \right) + f \left( {2x + y \over 3} \right) f(y) = f \left( {x + 2y \over 3} \right)^2 + f \left( {2x + y \over 3} \right)^2 \] for all \(x,y\) in \(\mathbb{R}\), and \[ f(x) = Ae^{Bx} \] for some positive constants \(A,B\) and for all \(x \in \mathbb{R}\). An open problem is posed.

MSC:

39B22 Functional equations for real functions

Biographic References:

Grégoire de Saint-Vincent
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References:

[1] Aczél, J., Lectures on Functional Equations and Their Applications, Academic Press, New York-London, 1966. · Zbl 0139.09301
[2] Aczel, J. and Dhombres, J., Functional Equations in Several Variables. Cambridge University Press, Cambridge, 1989.
[3] De Saint-Vincent, G., Opus geometricum quadratural circuli et sectionum coni. Problema Austriacum. Plus ultra quadratura circuli. Antwerpen, 1647.
[4] De Sarasa, A., Solutio problematis a R.P. Marino Merserno minimo propositi. Antwerper, 1649.
[5] Eves, H., An Introduction to the History of Mathematics, Holt, Rinehart and Winston, New York, 1976. · Zbl 0377.01001
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