I need to upper bound some complicated expressions involving binomial distributions:
Let $X \sim \mathrm{Binomial}(n,\frac{1}{2})$.
I want to find $a,b,c,m$ such that for $Y \sim \mathrm{Binomial}(m,\frac{1}{2})$.
$\forall x \in \{0,\ldots,n\} : \Pr[X=x]\Pr[X \le x] \leq a\Pr[Y=bx+c]$
and this upper bound is tight in some sense. I am loose in the definition of tight on purpose, since I am not completely aware of what I can hope for.
Does there exist general results about problems like this?