  Let $f$ be a smooth real function with strictly monotone first $k$
derivatives. We show that for a finite set $A$, with $|A+A|\leq K|A|$,
$|2^kf(A)-(2^k-1)f(A)|\gg_k |A|^{k+1-o(1)}/K^{O_k(1)}$. 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 $A \cdots A$.
