Hausdorff moment sequences and hypergeometric functions

Pólya in 1926 showed that the hypergeometric function $F(z)={\text{null}}_2{F}_1(a,b;c;z)$ has a totally monotone sequence as its coefficients; that is, $F$ is the generating function of a Hausdorff moment sequence, when $0\le a\le 1$ and $0\le b\le c\mathrm{.}$ In this paper, we give a complete characterization of such hypergeometric functions $F$ in terms of complex parameters $a,b,c\mathrm{.}$ To this end, we study the class of general properties of generating functions of Hausdorff moment sequences and, in particular, we provide a sufficient condition for the class by making use of a Phragmèn-Lindelöf type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike.