If $A+A$ is small then $AAA$ is superquadratic

This note proves that there exists positive constants $c_1$ and $c_2$ such that for all finite $A\subset \mathbb{R}$ with $\mathrm{\mid }A+A\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{1+c_1}$ we have $\mathrm{\mid }AAA\mathrm{\mid }\gg \mathrm{\mid }A{\mathrm{\mid }}^{2+c_2}$.