Given the parameters $ \frac{1}{2} < \theta < 1 , 0 < \rho < 1 , $ where $ N $ is a positive integer, and $ n $ is a non-negative integer such that $ n \leq N $, I am interested in determining when the following inequality holds:
$$[\theta^2 \cdot (2\theta\rho - 2\theta - 2\rho + 1) \cdot (1 - \theta)^{N-1} \cdot (1 - \rho)^{N-1} \cdot (\rho + \theta(1 - \rho))^{N-1}+ (1 - \theta)^2 \cdot (2\theta\rho - 2\theta + 1) \cdot \theta^{N-1} \cdot (1 - \rho)^{N-1} \cdot (1 - \theta(1 - \rho))^{N - 1}] \geq [(2\theta - 1) \cdot \theta^{n+1} \cdot (1 - \theta)^{n+1} \cdot (1 - \rho)^{2n+1} \cdot (1 - \theta(1 - \rho))^{N-n-1} \cdot (\rho + \theta(1 - \rho))^{N-n-1}+ (2\theta - 1) \cdot \theta^{N-n} \cdot (1 - \theta)^{N-n} \cdot (1 - \rho)^{2N-2n-1} \cdot (1 - \theta(1 - \rho))^n \cdot (\rho + \theta(1 - \rho))^n]$$
I have numerically verified that the inequality holds for many examples, but I also found cases where it does not hold. However, I am looking for an exact solution set that characterizes when this inequality is true.
How can I determine the solution set for this inequality given the constraints on $\theta$, $\rho$, $N$, and $n$?
Any guidance or references to relevant methods would be greatly appreciated.
