A convexity theorem for multiplicative functions

Abstract

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function \({g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}\), the function \({({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}\), is convex if β ≥ 0 and α ≥ β1 + ··· + βn. We also provide further generalization to functions of the form \({({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} }\) with the fk concave, positively homogeneous and nonnegative on their domains.