What is an easy way to show that for positive integers $i,n$, a real $p \in (\frac12,1)$ and $\epsilon \in [0,p]$,
$p^i(1-p)^{n-i} \geq (p-\epsilon)^i(1-(p-\epsilon))^{n-i}.$
(I have a complicated way, where I first show that the left hand side is bigger when i = n/2 and then increasing i can only make the left hand side bigger. But is there some well known inequality which lets me formulate this shorter?)