We describe a method for estimating the special function <^s> , in the complex cut plane $A = {\mathbf{C}}\backslash ³eft( { - \infty ,0} \right]$, with a Stieltjes transform, which implies that the function <^s> is \textit{logarithmically completely monotonic}. To be complete, we find a nearly exact integral representation. At the end, we also establish that $1 \mathord{³eft/ {\vphantom {1 }} \right. \kern-\nulldelimiterspace} \mbox{<^s>} ³eft( x \right)$ is a complete Bernstein function and we give the representation formula which is analogous to the L\'evy-Khinchin formula.