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Do you have any information if the completely monotonic function of $n$-variables function has been defined yet?

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    $\begingroup$ Have a look at arxiv.org/abs/1301.2449 -- it lists many references too. $\endgroup$
    – Suvrit
    Commented Dec 16, 2016 at 20:44

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You may like to look into Ressel paper here. If you don't have access then check out his presentation slides.

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