Complete Bernstein functions and real powers of P-matrices

Abstract

We classify the nonzero complete Bernstein functions whose matrix functions preserve every real P-matrix in every order. They are precisely $f(z)=a+bz$, with $a,b\ge0$ and $a+b>0$, and $f(z)=a+cz/(z+s)$, with $a\ge0$ and $c,s>0$. For each fixed order $n\ge4$, the preservers vanishing at zero are exactly $cz$ and $cz/(z+s)$; every other such function admits a strictly totally positive counterexample. For a finite positive measure $μ$ on $(0,\infty)$, put $g(z)=\int z/(z+s)\,μ(ds)$ and $M_t=\int s^t\,μ(ds)$. In every fixed order $n\ge4$, a finite constant shift restores preservation on all real P-matrices if and only if $M_{r_n}$ and $M_{-r_n}$ are finite, where $r_n=\lfloor n/2\rfloor-1$. Every $a>c_n\sqrt{M_{r_n}M_{-r_n}}$ is sufficient, with $c_n$ depending only on $n$. A complete Bernstein function vanishing at zero admits a possibly different repairing constant in every order exactly when it is linear, or is a pure kernel mixture with all positive and negative integer moments finite. For principal real powers, the preserving exponents are $\mathbb R$ in order one, $[-1,1]$ in orders two and three, and $\{-1,0,1\}$ in every higher order. An explicit rational determinant-one totally positive family gives negative principal minors uniformly on prescribed compact subsets of $(-1,0)\cup(0,1)$.