## Probability problem: upper bounds on binomial CDF

1. The problem statement, all variables and given/known data
Hi all, just a quick question here - the setup is as follows: X is a random variable, $X \sim \operatorname{Bin}(m,p)$ where $p=2^{-\sqrt{\log n}}(\log n)^2$ and $m \geq 2^{\sqrt{\log n}}c$ for constants c, n (n "large" here). I wish to show that $\mathbb{P}(X < c) \leq e^{-(\log n)^2 c/3}$. I've been told to use "Chernoff-esque bounds" here; however, after teaching myself a little about Chernoff bounds I haven't found a way to make this work - I can see that the multiplicative form could be useful but I haven't yet figured out how to translate the bounds which I've found online into a workable form for this problem.

I'm told observing the fact that $\mu = \mathbb{E}(X) \geq (\log n)^2 c = \mu '$ should also help, so I suspect maybe what we really need is to show is $\mathbb{P}(X < c) = \mathbb{P}(X < \frac{\mu'}{(\log n) ^2}) < \mathbb{P}(X < \frac{\mu}{(\log n) ^2}) \leq ^{(*)} e^{- \mu / 3} \leq e^{-\mu ' /3}$ but as I said, no luck so far since I can't prove step $(*)$ if indeed that is the way to do it. I suspect that this result only needs a few lines of work once you have the bound you require from Chernoff, so if anyone could show me how to do this I'd be very grateful! -S
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 No thoughts anyone? Sorry to bump but I could really use some help with this, any thoughts at all please respond!

 Tags inequality, probability