According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function:

$f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$

is a Bernstein function, meaning that its 1st, 3rd, 5th, ... derivatives are positive and its 2nd, 4th, .... derivatives are negative. (Equivalently, its derivative is a completely monotone function).

The closest citation in literature I could find is: https://link.springer.com/article/10.1007/s40590-016-0085-y?shared-article-renderer#Sec10, which proves it for 2 variables. Is this true or known in the general case for $n$ such variables?

  • 2
    $\begingroup$ Yes: the sum $g(t)$ of $1/(a_i + t)$ is a Stieltjes function (it satisfies $g(t) \geqslant 0$ for $t > 0$ and $\Im g(z) \leqslant 0$ when $\Im z > 0$), and thus $1 / g(t)$ is a (complete) Bernstein function. See, for example, the excellent book Berstein functions: Theory and Applications by Schilling, Song and Vondraček. $\endgroup$ Dec 14, 2019 at 10:44
  • $\begingroup$ What is a complete Bernstein function? $\endgroup$ Dec 14, 2019 at 10:48
  • 2
    $\begingroup$ One of the many equivalent definitions is: $f(t) \geqslant 0$ for $t > 0$ and $\Im f(z) \geqslant 0$ when $\Im z > 0$ (and $f$ holomorphic in $\mathbb{C} \setminus (-\infty, 0]$). Plus, of course, every complete Bernstein function is a Bernstein function. $\endgroup$ Dec 14, 2019 at 11:01


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