  A domain $G\subset \bar\mathbb C$ is the domain of holomorphy of the
generating function of a P\'olya frequency sequence of order $r$ if and only if
it satisfies the following conditions: (A) $G$ contains the point $z=0$, (B)
$G$ is symmetric with respect to the real axis, (C) $T=dist(0,\partial G)\in
\partial G$.
