Inequality of sizes of finite sets

Question

For a set $A\subset \mathbb N$, define $$A+c\cdot A=\{a+c\cdot a'|a,a'\in A\}$$

Is it true that for every such finite set $A$ $$\frac{|A+2A|}{|A|}\leq\left(\frac{|A+A|}{|A|}\right)^3$$?

Answer 1

This is indeed true: we have $2A \subset A+A$, hence $$ |A+2A| \leq |A+A+A| \leq \sigma(A)^3 |A|, $$ where $\sigma(A)=\frac{|A+A|}{|A|}$. The last inequality follows from the famous Plünnecke inequality (for instance, see here). The result holds for any finite subset $A$ of any additive group.