I'm trying to understand Ramanujan's proof of Bertrand's postulate, but I don't get the step in which it says
But is easy to see that
$\log\Gamma(x) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2}) \le \log[x]! - 2\log[\frac{1}{2}x]! \le \log\Gamma(x+1) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2})$
Then, applying Stirling's approximation, Ramanujan gets to:
$\log[x]! - 2\log[\frac{1}{2}x]! < \frac{3}{4}x$ if $x > 0$
and
$\log[x]! - 2\log[\frac{1}{2}x]! > \frac{2}{3}x$ if $x > 300$
How does he use Stirling formula to get those inequalities?
I know this have been asked before, but I'm trying to find another explanation because I'm not familiar with the method used by the user who answered.
I hope someone can help me understand.