I've encountered the following inequality which I am not able to prove but pretty certain that it is true: $$\Gamma\Big(x+\frac{1}{2}\Big)^2 < x\Gamma(x)^2$$ This should be true for $x\in\mathbb{R}^+$. However, I only need this for $x\in\mathbb{Q}^{+}$ or even $x=\frac{n}{4}$ for $n\in\mathbb{N}$, so I can prove $$\Gamma\Big(\frac{k+7}{4}\Big)^2 < \frac{k+5}{4}\Gamma\Big(\frac{k+5}{4}\Big)^2$$ and similar expressions for $k\in\mathbb{N}$.
In general, I have trouble with terms involving $\Gamma\Big(x+\frac{1}{2}\Big)$. I am aware of Gautschi's Inequality, but find it not quite sharp enough for small $x$. So I would also be thankful for any information on inequalities for $\Gamma\Big(x+\frac{1}{2}\Big)$.
Thanks, Nils
Gautschi's Inequality works just fine. Take $s=1/2$, then $$x^{1/2} < \frac{{\Gamma (x + 1)}}{{\Gamma (x + 1/2)}} = \frac{{x\Gamma (x)}}{{\Gamma (x + 1/2)}}$$ for $x>0$. Re-arrange to get $$\Gamma (x + 1/2) < x^{1/2} \Gamma (x)$$ for $x>0$.