On functions and inverses, both positive, decreasing and convex: And Stieltjes functions

Abstract

Abstract Any function from onto which is decreasing and convex has an inverse which is positive, decreasing and convex. When has some form of generalized convexity we determine additional convexity properties inherited by . When is positive, decreasing and -convex, its inverse is -convex. Related properties which pertain when is a Stieltjes function are developed. The results are illustrated with the Stieltjes function via a transcendental equation.