Let $x\geq 1$ a real number and $n\geq 3$ a natural number then we have : $$\frac{\Gamma(x+n)}{\Gamma(x)}\leq \Bigg(\frac{nx+\Big(\frac{n(n-1)}{2}\Big)}{\frac{n}{2}}-\Bigg(x^x(x+1)^{x+1}(x+2)^{x+2}\cdots(x+n-1)^{x+n-1}\Bigg)^{\frac{1}{nx+\Big(\frac{n(n-1)}{2}\Big)}}\Bigg)^{n}$$
It's a conjecture but first for me :$\frac{\Gamma(x+n)}{\Gamma(x)}=x(x+1)\cdots(x+n-1)$.So if there are problems of notations feel free to correct me . I have tested for $n\leq 25$. If it's true I find this inequality beautiful . I conjecture also that the difference tends to $0$ when $x$ tends to infinity . I have tried Gautschi's inequality and more without success .
If you have an idea to solve it...
Any helps is greatly appreciated .
Thanks in advance for all your contributions .