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

Any function f from (0, infinity) onto (0, infinity) which is decreasing and convex has an inverse g which is positive, decreasing and convex. When f has some form of generalized convexity we determine additional convexity properties inherited by g. When f is positive, decreasing and p; q-convex, its inverse g is q; p-convex. Related properties which pertain when f is a Stieltjes function are developed. The results are illustrated with the Stieltjes function f (x) - arctano(1/sqrt(x)/sqrt(x) via a transcen-dental equation.