Higher convexity and iterated sum sets

Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $\mathrm{\mid }A+A\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$, $\mathrm{\mid }{2}^{k}f(A)-(2^k-1)f(A)\mathrm{\mid }{\gg }_k\mathrm{\mid }A{\mathrm{\mid }}^{k+1-o(1)}\mathrm{/}{K}^{O_k(1)}$. We deduce several new sum-product type implications, e.g. that $A+A$ being small implies unbounded growth for a many enough times iterated product set $A\cdots A$.