It is well known that a function f of the real variable x is convex if and only if (x, y) → yf(y–1x), y>0 is convex. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In this paper, we obtain a conjugacy formula which gives rise, as a corollary, to a new rule for generating new convex functions from old ones. In particular, it allows to extend the aforementioned property to functions of the form (x, y) → g(y)f(g(y)–1x) and provides a new tool for the study of the multiplicative potential and penalty functions.
