I am looking for bounds (upper and lower) for the falling factorial $n^{\underline{k}} = \frac{n!}{(n-k)!}$ that are valid for all $n \geq k$ and either coencides with or is close to the robbins bounds:
$\sqrt{2 \pi} n^{n+\frac{1}{2}} e^{-n} e^{\frac{1}{12n+1}} < n! < \sqrt{2 \pi} n^{n+\frac{1}{2}} e^{-n} e^{\frac{1}{12n}}$
I've spent a long time searching, and all the bounds that I know either only work for $n > k$ or have a very weak bound when $n = k$. I have also tried to do this myself using the Robbins bounds to bound $n!$ and $(n-k)!$, but this is only valid when $n>k$
Thanks!