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)kis completely monotone (resp. alternating)? This approach constitutes a kind of converse to Hausdorff’s moment characterization theorem in the context of completely monotone sequences. Keywords: Completely monotone functions, Completely monotone sequences, Bernstein functions, Completely alternating functions, Completely alternating sequences, Hausdorff moment problem, Hausdorff moment sequences, Self-decomposability, 2010 Mathematics Subject Classification: 30E05, 44A10, 44A60, 47A57, 60E05, 60E07, 60B10
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