Domains of holomorphy of generating functions of Polya frequency sequences of finite order

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, © $T=dist(0,\partial G)\in \partial G$.