A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : R-m -> [0, infinity], the function (x, y) bar right arrow g(x)(1+alpha) y(-beta), y > 0, is convex if beta >= 0 and alpha >= beta(1) + ... + beta(n). We also provide further generalization to functions of the form (x, y(1), ..., y(n)) bar right arrow g(x)(1+alpha) f(1)(y(1))(-beta 1) center dot center dot center dot f(1)(y(1))(-beta n) with the f(k) concave, positively homogeneous and nonnegative on their domains.