## Duplicating the cube and functional equations.(English)Zbl 0829.39008

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.

### MSC:

 39B62 Functional inequalities, including subadditivity, convexity, etc. 39B22 Functional equations for real functions
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### References:

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