Given $q \in \mathbb N$, $q\geq 2$ is it true that \begin{equation*} \sum _{i=0}^a (q-1)^i\binom {n}{i} \leq q^{H_q(a/n)n}? \end{equation*} Here $H_q(x) = x\log _q(1/x) + (1-x)\log _q(1/(1-x))$ is the base $q$ entropy function. Also $a$ might need to satisfy some sensible bound like $a \leq (q-1)n/q$. I know this is true when $q = 2$, but it seems pretty believable for general $q$. It would give a tidier bound for something I'm writing.
Thanks!