The principal inverse function of the function gamme is denoted by $Γ⁻¹(x)$. See the paper: https://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2/S0002-9939-2011-11023-2.pdf

$Γ⁻¹(x)$ is an increasing and concave function defined on $(0.8856,+∞)$.

I am asking about the existence of an interval $I⊂ (0.8856,+∞)$ such that the derivative $(Γ⁻¹(x))′$ is greater than $1$ for all $x$ in $I$.

The principal inverse function of the gamma function is denoted by $\Gamma^{-1}$. See the paper: Uchiyama - The principal inverse of the gamma function.

$\Gamma^{-1}$ 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 $I$.