The principal inverse function of the gamma function is denoted by $\Gamma^{-1}$. See the paper: [Uchiyama - The principal inverse of the gamma function](https://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2).


$\Gamma^{-1}$ is an increasing and concave function defined on $(0.8856,+\infty)$.


I am asking about the existence of an interval $I\subset (0.8856,+∞)$ such that the derivative $(\Gamma^{-1})’(x)$ is greater than $1$ for all $x$ in $I$.



The existence of an interval $I\subset (0.8856,+\infty)$ such that the derivative $(\Gamma^{-1})’(x)$ is greater than $1$
Am I missing something? $(\Gamma^{-1})'(y) = \frac{1}{\Gamma '(x)} = \frac{1}{\Gamma(x) \cdot \psi(x)}$ for $y = \Gamma(x)$. The existence of $I$ now immediately follows from the continuity of $(\Gamma^{-1})'$.


