Growth beyond exponent $3\mathrm{/}2$ for convexity and iterated sum sets

We prove that the bound $$\max \{\mathrm{\mid }16A\mathrm{\mid },\mathrm{\mid }16f(A)\mathrm{\mid }\}{\gg }_m\mathrm{\mid }A{\mathrm{\mid }}^{\frac{3}{2}+\frac{1}{162}}$$ holds for any polynomial $f$ with degree $m\ge 2$ and any finite $A\subset \mathbb{R}$. This shows that the classical Jarník obstruction to growth beyond exponent $3\mathrm{/}2$, which occurs for general strictly convex functions, cannot occur for polynomial functions.