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?

Berstein functions: Theory and Applicationsby Schilling, Song and Vondraček. $\endgroup$