Hi guys, I'm reading a research paper and they have this thing: Given this inequality $\alpha^2 w_1 + (1-\alpha)^2 w_3 \leq (\alpha^2 + (1-\alpha)^2)w_2, \alpha \in [0, 1]$ and this relation $w_1 > w_2 > w_3$. They claim that there exists $\alpha^* \in [0, 1]$ such that the inequality holds for all $\alpha < \alpha^*$. The problem is paper provides no proof (perhaps it's too easy!?) and I can't prove this. I think the claim is correct because the paper has been published, but it bothers me that I can't convince myself. Can you gimme some ideas how this is correct?
