On a functional operation generating convex functions, part 1:: Duality

The function f Delta g : (x, y) bar right arrow g( y) f ( x/ g( y)), y is an element of dom g, is jointly convex provided f is convex and nonpositive at the origin and provided g is concave and nonnegative on its effective domain. Its convex conjugate combines the convex conjugates of f and - g by means of the same composition law. The effective domain of f Delta g is then studied, which will prove to be useful in Part 2 of this paper (algebraic properties, Ref. 1).