Let $a_1,\dots,a_k$ be non-negative and sum to $1$ and let $x_1,y_1,\dots,x_k,y_k$ be positive. Then is it true that $$\prod_{i=1}^kx_i^{\alpha_i}+\prod_{i=1}^ky_i^{\alpha_i}\leq\sum_{i=1}^k(x_i+y_i)^{\alpha_i}?$$
This is example $1.2.3$ in Convex Analysis and Minimization Algorithms $I$ by Hiriart-Urruty and Lemarachal.
Consider what happens when all the $x_i$'s and $y_i$'s equal some value $x$ and the $\alpha_i$'s equal $1/k$. Then the claim is that $$ 2x=2\prod_{i=1}^kx^{1/k}\leqslant \sum_{i=1}^k(2x)^{1/k}=k(2x)^{1/k}. $$
But if $x>(2^{1/k - 1} k)^{1/(1 - 1/k)}$ this is false.