inequality - Lower and upper bound of the Stirling's approximation

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?