Let $X \sim \text{Bin}(n,p)$. Wikipedia claims 
$$\mathbf P[X \leq (p-\epsilon)n ] \leq e^{ - 2 \epsilon^2 n}.$$
This follows from Hoeffding's inequality (<https://en.wikipedia.org/wiki/Hoeffding%27s_inequality#General_case>).


Is this tight? Could a more clever application of Hoeffding, or perhaps Chernoff bounds yield a larger exponent? I am trying to optimize a constant in a calculation and would like the best bound possible.



Tight binomial left tail bound
This is tight, at least when $p=\frac12$. You simply need to approximate $\log\big({n \choose \frac{n}{2}-\varepsilon n} \frac1{2^{n}}\big)$ using Stirling's formula and you'll see that the leading coefficient is indeed $-2\varepsilon^2 n$.


