Use the conventions $\Pi(x)=\Gamma(x+1)$ and $\frac{d^n}{dx^n}\Pi(x)=\Pi^{(n)}(x)$ henceforth.
I am trying to evaluate for what real $\color{blue}{x}>2$ it holds that $$\frac{\Pi^{(\color{red}{n})}(\color{blue}{x})}{\Pi(\color{red}{n})}>1$$
The only helpful information about expanding the ratio is a function defined recursively as $$P(n,x)=\frac{d \ln[\Pi(x)]}{dx} \,P(n-1,x)+\left.\frac{\partial P(n,x)}{\partial x}\right|_{(n-1,x)} = \frac{\Pi^{(n)}(x)}{\Pi(x)}$$ given that $P(0,u)=1$ (very special thanks to @SimplyBeautifulArt for helping me with this).
I have absolutely no clue what to do from here. How should I proceed?
(Please note that this isn’t a homework problem, just a small piece of a more thorough investigation that I am conducting independently.)