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.