Is it true that for $0<\alpha<1$ and $x_1,\dots,x_n,y_1,\dots,y_n\in\mathbb{R}$ we have the inequality:
$$ \log \sum_{i=1}^n e^{\alpha x_i+(1-\alpha) y_i} \leq \alpha\, \log \sum_{i=1}^n e^{x_i}+(1-\alpha) \, \log \sum_{i=1}^n e^{y_i} $$
The furthest I could prove is that
$$ \log \sum_{i=1}^n e^{\alpha x_i+(1-\alpha) y_i} \leq \alpha\, \log \sum_{i=1}^n e^{x_i}+(1-\alpha) \, \log \sum_{i=1}^n e^{y_i} + \log n $$