Optimal iterated-sum growth for images of k-convex functions

Prove that for every finite set A\subset R, every k\in\mathbb{N}, and every k-convex function f, the estimate |2^k f(A-A)-(2^k-1)f(A-A)|\gg |A|^{k+1} holds.

Background

The paper discusses existing lower bounds for additive growth of iterated sumsets of function images. It notes that Bradshaw conjectured an analogue of the sharp higher-convexity estimate for sets, namely a bound with exponent k+1 for the iterated sumset of f(A-A). The paper proves weaker bounds for a broad analytic class, so the conjectured estimate remains unresolved.

References

Indeed, Bradshaw conjectures that for any $k$-convex function $f$, $$|2kf(A-A)-(2k-1)f(A-A)| \gg |A|{k+1}$$ should hold for all finite subsets $A \subset R$ and $k \in \mathbb{N}$.

— Additive growth amongst images of linearly independent analytic functions  (2503.03690 - Mansfield, 5 Mar 2025) in Section 1, subsection “Growth for sum sets of convex functions”