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.


39B22 Functional equations for real functions

Biographic References:

Grégoire de Saint-Vincent
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