Inequalities for binomial coefficients

In this paper we prove several inequalities for binomial coefficients. For instance, if $k$ and $n$ are positive integers such that $n\ge 400$ and $[\frac{n}{5}]\le k\le [\frac{n}{2}]$, where $[x]$ is the greatest integer not exceeding $x$, then $$\left(\genfrac{}{}{0px}{}{n}{k}\right)<(1-\frac{5(k-[\text{f}n5])}{6n^2})\frac{n^{n-\text{f}12}}{k^k(n-k{)}^{n-k}}\mathrm{.}$$