The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function

Abstract

In the paper, by the Cauchy integral formula in the theory of complex functions, an integral representation for the reciprocal of the weighted geometric mean of many positive numbers is established. As a result, the reciprocal of the weighted geometric mean of many positive numbers is verified to be a Stieltjes function and, consequently, a (logarithmically) completely monotonic function. Finally, as applications of the integral representation, in the form of remarks, several integral formulas for a kind of improper integrals are derived, an alternative proof of the famous inequality between the weighted arithmetic and geometric means is supplied, and two explicit formulas for the large Schröder numbers are discovered.