I have a conjecture which has been verified extensively by simulation.
The conjecture is as follows:
$\forall t \in [0, 1], \alpha \in [0,1]$, and positive real sequences $\{p\}_{i:1,\dots,n}, $, $\{q\}_{i:1,\dots,n}, $, it holds that $$\sum_{i,j} (t p_i p_j + \bar{t} q_i q_j)^{1+\alpha} \geq \sum_{i,j} (t p_i + \bar{t} q_i)^{1+\alpha} (t p_j + \bar{t} q_j)^{1+\alpha},$$ where $\bar{t} = 1-t$.
For the case $\alpha = 0$ and $\alpha = 1$, it's easy to prove the conjecture, by expanding the terms and using convexity and Jensen's inequality. For example, when $\alpha = 0$, one can use the fact that $(\sum_i x_i)^2$ is convex.
However, I don't have any clue of how to prove it for the more general case where $\alpha \in (0,1)$.
Does anyone have any idea of how to prove the general case (possibly by leveraging the convexity of the terms)? Thanks in advance!