Minimize over $m$ the expression:
$E(m) = \sum_{i = 0}^{n - 1} {m \choose i} \cdot \left[ c \cdot \left(\frac{N}{m} \right)^2 \right]^i \cdot \left[1 - c \cdot \left(\frac{N}{m} \right)^2 \right]^{m - i}$
where $c > 0$ is a constant such that $1 - c \cdot \left(\frac{N}{m} \right)^2 > 0$ and $n \leq m \leq N$.
The problem is to minimize $E(m)$ over integer $m$ such that $n \leq m \leq N$.
Because of the binomial coefficients, using the derivative does not seem to be an option and I am not sure what types of inequalities to use that could simplify the problem.
If you have any suggestion about how to tackle this problem I would much appreciate.