  We prove the converse to a result of Karlin [Trans. AMS 1964], and also
strengthen his result and two results of Schoenberg [Ann. of Math. 1955]. One
of the latter results concerns zeros of Laplace transforms of multiply positive
functions. The other results study which powers $\alpha$ of two specific
kernels are totally non-negative of order $p\geq 2$ (denoted TN$_p$); both
authors showed this happens for $\alpha\geq p-2$, and Schoenberg proved that it
does not for $\alpha<p-2$. We show more strongly that for every $p \times p$
submatrix of either kernel, up to a 'shift', its $\alpha$th power is totally
positive of order $p$ (TP$_p$) for every $\alpha > p-2$, and is not TN$_p$ for
every non-integer $\alpha\in(0,p-2)$. In particular, these results reveal
'critical exponent' phenomena in total positivity. We also prove the converse
to a 1968 result of Karlin, revealing yet another critical exponent phenomenon
- for Laplace transforms of all Polya Frequency (PF) functions.
  We further classify the powers preserving all TN$_p$ Hankel kernels on
intervals, and isolate individual kernels encoding these powers. We then
transfer results on preservers by Polya-Szego (1925), Loewner/Horn (1969), and
Khare-Tao (in press), from positive matrices to Hankel TN$_p$ kernels. Another
application constructs individual matrices encoding the Loewner convex powers.
This complements Jain's results (2020) for Loewner positivity, which we
strengthen to total positivity. Remarkably, these (strengthened) results of
Jain, those of Schoenberg and Karlin, the latter's converse, and the above
Hankel kernels all arise from a single symmetric rank-two kernel and its
powers: $\max(1+xy,0)$.
  We also provide a novel characterization of PF functions and sequences of
order $p\geq 3$, following Schoenberg's 1951 result for $p=2$. We correct a
small gap in his paper, in the classification of discontinuous PF functions.
