Inverses of Gamma Functions

Euler’s Gamma function Γ either increases or decreases on intervals between two consecutive critical points. The inverse of Γ on intervals of increase is shown to have an extension to a Pick function, and similar results are given on the intervals of decrease, thereby answering a question by Uchiyama. The corresponding integral representations are described. Similar results are obtained for a class of entire functions of genus 2, and, in particular, integral representations for the double gamma function and the G -function of Barnes are found.