Duplicating the cube and functional equations
The authors make some interesting contributions to the study of the system of functional equations \[ f \bigl( px + (1 - p)y \bigr) f \bigl( (1 - p)x + py \bigr) = f(x) \cdot f(y), \tag{1} \] \[ f(x)f \bigl( px + (1 - p)y \bigr) + f(y)f \bigl( (1 - p)x + py \bigr) = f \bigl( px + (1 - p)y \bigr)^2 + f \bigl( (1 - p)x + py \bigr)^2. \tag{2} \] These equations were introduced by C. Alsina and J. L. Garcia-Roig in relation with the study of some results of G. de Saint Vincent related to the classical Greek problem of duplicating the cube. In this paper equations (1) and (2) are solved (jointly or separated) under different regularity conditions and for different values of \(p\) \((p \neq 1/3;\;p = 1/3;\;p \in Q; \ldots)\). In the restricted domain \(y = 2x\) the equations are investigated with detail.
scientific article; zbMATH DE number 785177
scientific article