New characterizations of completely monotone functions and Bernstein functions, a converse to Hausdorff’s moment characterization theorem

We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on N0. We give a complete answer to the following question: Can we affirm that a function f is completely monotone (resp. a Bernstein function) if we know that the sequence f(k)k is completely monotone (resp. alternating)? This approach constitutes a kind of converse to Hausdorff’s moment characterization theorem in the context of completely monotone sequences.