I have this function $$f(\lambda)= \sum_i p_i\cdot\frac{\lambda_i^2}{w_i^2} - \frac{\left(\sum_i g_i\cdot \lambda_i\right)^2}{\left(\sum_i g_i\cdot w_i\right)^2},$$ where $p_i, g_i$ and $w_i$ are probabilities that sum to $1$, $\lambda_i's$ are greater than $0$.
I conjecture this function is always greater or equal to zero, but could not prove it. Any suggestions?
Edit: As @Youem showed, this function is not always greater than 0. However, this new function does $$f(\lambda)= \sum_{i=1}^N \frac{g_i\lambda_i^2}{w_i \sum_{k=1}^Ng_k\cdot w_k} - \frac{\left(\sum_i g_i\cdot \lambda_i\right)^2}{\left(\sum_i g_i\cdot w_i\right)^2}.$$
This is not true in general, for example $n=2$, $p_1 = 1$, $p_2 = 0$, $g_1 = 0$, $g_2 = 1$, $w_1 = 1/2$, $w_2 = 1/2$, $\lambda_1 = 1$ and $\lambda_2 = 2$
$$f(\lambda) = 4 - 4\times 4 = -12$$