The following problem is somehow hidden in this recently asked question, but I believe that it deserves to be asked explicitly.

Is it true that for any finite set $A$ of real numbers, and any real $\lambda\notin\{0,-1\}$, one has $$ |A+\lambda A| \ge |A+A| $$ (where $\lambda A=\{\lambda a\colon a\in A\}$ is the dilate of $A$ by the factor $\lambda$, and $A+B=\{a+b\colon a\in A,\ b\in B\}$ is the sumset of $A$ and $B$)?

The energy version seems equally interesting to me. Let $T_A(\lambda)$ denote the number of representations of $\lambda$ in the form $\frac{a_1-a_2}{a_3-a_4}$ with $a_1,a_2,a_3,a_4\in A$. It is easily seen that $T_A(-\lambda)=T_A(\lambda)$ and $T_A(\lambda)<T_A(0)=|A|^2(|A|-1)$ for any real $\lambda\ne 0$.

Is it true that for any finite set $A$ of real numbers, and any real $\lambda\ne 0$, one has $$ T_A(\lambda) \le T_A(1) ? $$

I have not done any computations, so maybe it is possible to find a counterexample just by a computer search.

Both questions have now received nice and exhaustive answers thanks to Boris Bukh, Kevin Costello, and Terry Tao (who has actually answered even before the question got asked). Unfortunately, I cannot accept more than one answer; so, I am accepting only that which, for some reason, got less votes.