currently I am stocked at proving an inequality. Since it is very specific I don't want to go into detail, but it has a form like this:
$$ \sum_{i=0}^{n} \binom{n}{i} f(i)^{i} g(i)^{n-i}$$
where f and g are polynomial functions mapping into $(0,1)$. Sure it is possible to bound g and f and then apply the binomial theorem, but the bound I receive is not good enough.
Are there any other known inequalities I can apply, which are stronger than the method above, without using an upper bound for $ f(i)$ and $g(i)$?
Thank you
EDIT
Exact form: $$ \sum_{i=0}^{n} \binom{n}{i} \Big( 1-\frac{n-i}{n} \Big)^i \Big( \frac{i}{n^2}\Big)^{n-i} $$