# zbMATH — the first resource for mathematics

On the functional equations $$\phi (x)=\alpha \phi (\alpha x)+(1- \alpha)\phi (1-(1-\alpha)x)$$. (English) Zbl 0608.39003
The authors deal with the functional equation $$\phi (x)=\alpha \phi (\alpha x)+(1-\alpha)\phi (1-(1-\alpha)x)$$, $$x\in [0,1]$$, where $$\alpha\in (0,1)$$ is a fixed constant and $$\phi$$ :[0,1]$$\to {\mathbb{R}}$$. They construct the general solution of the equation starting from an arbitrary function $$\phi_ 0$$ defined on the interval [0,$$\alpha$$ ] and such that $$\phi_ 0(0)=\phi_ 0(\alpha)$$. Moreover it is proved that if a solution $$\phi$$ is either Riemann integrable or bounded and continuous at least at one point, then it is constant.
Reviewer: G.L.Forti
##### MSC:
 39B99 Functional equations and inequalities