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

In the paper, by the Cauchy integral formula in the theory of complex functions, an integral representation for the reciprocal of the geometric mean of many positive numbers is established. As a result, the reciprocal of the geometric mean of many positive numbers is verified to be a Stieltjes transform and, consequently, a (logarithmically) completely monotonic function.