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
Full Text:

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.