An upper bound for the function $x!$ using some well-know constant as $e$ or $\pi$

Solution 1:

All this work has been done solving with (more than nasty) radicals quartic equations generated for approximations.

Starting with a small detail (already reported by @River Li in comments) : the inequality $j(x) > \Gamma(x+1)$ does not hold for very small values of $x$ close to $0$. $$\Delta=j(0) - \Gamma(1)=\left(1-\frac{1}{e^3}\right)^{3\frac{ (\pi -3)}{\pi ^2}}-1 \quad < 0$$ So, if the inequality holds, it will be for $0 < x_* < x $.

Considering the function $$h(x)=j(x)-\Gamma(x+1)$$ its series expansion around $x=0$ gives a quartic polynomial in $\sqrt x$. Expanded to $O\left(x^{5/2}\right)$, $h(x)=0$ for $\color{blue}{x_0=0.0278927}$ (the "exact" solution being $\color{red}{x_0=0.0278222}$).

Expanding $h'(x)$ as a series around $x=1$ to $O\left((x-1)^{5}\right)$ shows a maximum value of $h(x)$ for $\color{blue}{x_1=0.760950}$ (the "exact" solution being $\color{red}{x_1=0.761132}$) and $\color{blue}{h(x_1)=0.0888744}$ (the "exact" value being the same).

Expanding $h'(x)$ as a series around $x=e$ to $O\left((x-e)^{5}\right)$ shows a minimum value of $h(x)$ for $\color{blue}{x_e=2.55947}$ and $\color{blue}{h(x_e)=0.0109763}$.

Similarly, expanding $h''(x)$ around $x=\frac \pi 2$ (this is close to $\frac {x_1+x_e}2$), the inflection is predicted at $\color{blue}{x_i=1.74822}$ (the "exact" value being $\color{red}{x_i=1.74814}$)

Reusing the first series expansion leads to $\color{blue}{x_*=0.0671382}$ (the "exact" value being $\color{red}{x_*=0.0662558}$)

A numerical minimization shows that $\color{red}{x_{\text{min}}=2.55956}$ and $\color{red} {h({x_{\text{min}}})=0.0109763}$