That's a very quick way to prove it, thanks! This proof is of particular interest to me because I was also aiming to prove \Gamma(\alpha) \geq \alpha^{\alpha+1}, which indeed is essentially equivalent to the inequality on Beta by using Stirling on both \Gamma(x\alpha) and \Gamma((x+1)\alpha).

That's a wonderful and clever proof, many thanks! I'm marking this as accepted (although @egs's one also proves the desired inequality) also because it came first.