Let $n$ be an integer and $p\in (0, 1)$. I am looking for an upper bound on $$\max_{0\leq i\leq n} \binom ni p^i(1-p)^{n-i}.$$ In other words, I want an upper bound on how much mass of a binomial distribution with parameters $n$ and $p$ can reside at any one point. By guessing that the maximum is obtained at $i=pn$ (which may be non-integer), Sterlings approximation yields an estimate of $O(1/\sqrt{np(1-p)})$. I could probably formalise this. However, it does not feel like a very clean argument.
Is there a more elegant solution?
The mode of a binomial distribution occurs at one of the following two values: $$\lfloor (n+1)p \rfloor, \quad \text{or} \quad \lceil (n+1)p \rceil - 1.$$ This may then be used to compute the desired upper bound. Note that since in all cases except $p = 1$, the choice $i = \lfloor (n+1) p \rfloor$ corresponds to a maximum value, your upper bound need not consider when the mode occurs across two possible values of $i$.