Let $q$ be a positive integer. Is it true there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of reals: $$\displaystyle A+qA\ge (q+1)AC_q\qquad (1)$$ I got this idea from the wellknown inequality $A+A\ge 2A1$, so I was thinking about the general case, but no idea about it. I experimented a bit with small values of $A$ and it seems to be true. This paves the road for a far more general inequality $$ kA+\ell A\ge (k+\ell)AC_{k,\ell}\qquad (2)$$ with $k,\ell$ integers and $\text{gcd}(k,\ell)=1$. So, an answer would be welcome. A solution of $(1)$ is the thing I ask for, $(2)$ is just a bonus. And I haven't checked $(2)$ for small values yet. Anyways proving or disproving them both will be helpful. Thanks a lot.

1$\begingroup$ There is no way (2) could hold without at least requiring $k$ and $l$ to be coprime: if $m>1$ and $A=\{1,\dots,n\}$, then $kmA+lmA=kA+lA\le(k+l)n<(km+lm)nC_{km,lm}$ for $n$ large enough. $\endgroup$ – Emil Jeřábek Jun 15 '14 at 15:43

$\begingroup$ Thank you so I am adding the condition in the question. I knew the second one was faulty somehow. $\endgroup$ – shadow10 Jun 15 '14 at 15:46
Inequality (2) is true provided $(k,\ell)=1$  this is a recent result of Balog and Shakan in 'On the sum of dilates of a set', http://arxiv.org/pdf/1311.0422.pdf.
They show that for any finite $A\subset\mathbb{Z}$ and integers $k,\ell$ such that $(k,\ell)=1$.
$$ \lvert kA+\ell A\rvert \geq (k+\ell)\lvert A\rvert  (k\ell)^{(k+\ell3)(k+\ell)+1}. $$