Complete Bernstein functions and real powers of P-matrices

@misc{xie2026complete,
  title = {{Complete Bernstein functions and real powers of P-matrices}},
  author = {Wei Xie},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00096},
  eprint = {2611.00096},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00096},
}

TY  - UNPB
TI  - Complete Bernstein functions and real powers of P-matrices
AU  - Wei Xie
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00096
UR  - https://arxiv.org/abs/2611.00096
AB  - 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≥0 and a+b>0, and f(z)=a+cz/(z+s), with a≥0 and c,s>0. For each fixed order n≥4, 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,∞), put g(z)=∫ z/(z+s) μ(ds) and M_t=∫ s^t μ(ds). In every fixed order n≥4, 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=⌊ n/2⌋−1. Every a>c_n√(M_r_nM_−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 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)∪(0,1).
ER  -