An inequality on sumsets. Given two finite sets $A,B\subset \mathbb{R}$, can we assert the inequality $$|A+B|^2\ge |A+A|\cdot|B+B|?$$
I tried to construct an injective function from $(A+A)\times (B+B)$ to $(A+B)^2$ but failed to make the function injective. 
It's a problem from my classmates, I'm not 100% sure if it's true. 

As noted in zhoraster's comment this is false even for $B=-A,$ see "An old question about sumsets and difference sets".
