Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity

This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: R k ,[0,1[ k ,N0‐the first with the inverse involution and the two others with the identical involution. Schoenberg’s theorem explains the possibility of constructing rotation invariant positive definite and conditionally negative definite functions on euclidean spaces via completely monotonic functions and Bernstein functions. It is therefore important to be able to decide complete monotonicity of a given function. We combine complete monotonicity with complex analysis via the relation to Stieltjes functions and Pick functions and we give a survey of the many interesting relations between these classes of functions and completely monotonic functions, logarithmically completely monotonic functions and Bernstein functions. In Section 6 it is proved that logx ( x) and 0 (x) are logarithmically completely monotonic (where ( x) = 0 (x)/( x)), and these results are new as far as we know. We end with a list of completely monotonic functions related to the Gamma function.