[Math] Lower and upper bound of the Stirling’s approximation

Question

Perhaps everybody has heard of the Stirling's approximation, namely:
$$ \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z $$
Thus (the very basic example):
$$ \Gamma\left(\frac{1}{2}z\right) \approx\sqrt{\frac{4\pi}{z}}\left(\sqrt{\frac{z}{2e}}\right)^{z}$$
My question is: how does one obtain the lower and upper bound for the Gamma function using the Stirling's approximation? I've heard that $$ \sqrt{\frac{4\pi}{z}}\left(\sqrt{\frac{z}{2e}}\right)^{\color{red}{z-1}}<\Gamma\left(\frac{1}{2}z\right)<\sqrt{\frac{4\pi}{z}}\left(\sqrt{\frac{z}{2e}}\right)^{\color{red}{z+1}} $$ is a very "ugly" and unproper way (moreover it doesn't always work). So what's the best way to obtain the lower and upper bound?

Answer 1

Hint

If you look here, you will find this very nice inequality $$\sqrt{2\pi}n^{n+\frac 12}e^{-n} \leq n! \leq en^{n+\frac 12}e^{-n}$$ which would give you good approximations of upper and lower bounds for the $\Gamma$ function.

For example $$\Gamma(12.3456)\approx 9.33280\times 10^7$$ while the above formulae give $\approx 9.26452\times 10^7$ and $\approx 1.00468\times 10^8$

$$\Gamma(123.456)\approx 8.85315\times 10^{203}$$ while the above formulae give $\approx 8.84713\times 10^{203}$ and $\approx 9.59416\times 10^{203}$