On some trigonometric functional inequalities. (English) Zbl 1010.39012

Daróczy, Zoltán (ed.) et al., Functional equations–results and advances. Dordrecht: Kluwer Academic Publishers. Adv. Math., Dordr. 3, 3-15 (2002).
The authors deal with stability results concerning d’Alemberts equation \[ f(x+ y)+ f(x- y)= 2f(x) f(y) \] and Wilson’s equation \[ f(x+ y) f(x-y)= f(x)^2- f(y)^2, \] where the target space is some commutative Banach algebra. They use their results to characterize the functions \(x\mapsto\cos(\alpha x)\) and \(x\mapsto \beta\sin(\alpha x)\), where \(\beta\in \mathbb{C}\setminus\{0\}\) and \(\alpha\in \mathbb{C}\setminus \mathbb{R}\), by some regularity properties and some functional inequalities.
For the entire collection see [Zbl 0983.00041].
Reviewer: J.Schwaiger (Graz)


39B62 Functional inequalities, including subadditivity, convexity, etc.
39B52 Functional equations for functions with more general domains and/or ranges
39B82 Stability, separation, extension, and related topics for functional equations