Maple seems to suggest that for any real $a\ge 1$ and positive integer $K$ and $n$ with $K\le n/(a+1)$ one has $$ a^n + na^{n-1} + \binom{n}{2}a^{n-2} +...+ \binom{n}{K}a^{n-K} \le a^{n-K} e^{nH(K/n)}, $$ where $H(x)=-x\log(x)-(1-x)\log(1-x)$ is the entropy function.

An essentially equivalent form is as follows. Suppose that $X\sim B(n,p)$, where $p<1/2$, and let $q:=1-p$. Then for any $\alpha\le 1$, $$ \mathsf{Pr}(X\le\alpha pn) \le p^{\alpha pn}q^{(1-\alpha p)n}e^{nH(\alpha p)}. $$

I believe this should be well-known (if at all true). Can anybody suggest a reference?