# A note on the convexity of the multiplicative potential and penalty functions

Marechal, Pierre (1999) A note on the convexity of the multiplicative potential and penalty functions. [Preprint]

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Postscript
It is well known that a function~$f$ of the real variable~$x$ is convex if and only if $(x,y)\to y f(y^{-1} x)$ is convex for $y>0$. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In Theorem 1, a rule for generating new convex functions from old ones is obtained. It generalizes the aforementioned property to functions of the form $(x,y)\to g(y) f(g(y)^{-1} x)$ and provides deeper insight into the structure of the multiplicative potential and penalty functions.