On generating functions of Hausdorff moment sequences

The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-infty,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$. Also we provide a simple analytic proof that for any real $p$ and $r$ with $p>0$, the Fuss-Catalan or Raney numbers $frac{r}{pn+r}binom{pn+r}{n}$, $n=0,1,ldots$ are the moments of a probability distribution on some interval $[0,tau]$ {if and only if} $pge1$ and $pge rge 0$. The same statement holds for the binomial coefficients $binom{pn+r-1}n$, $n=0,1,ldots$.