A better than exponent for iterated sums and products over

In this paper, we prove that the bound \begin{equation*}\max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}}\end{equation*} holds for all $A \subset \mathbb R$ , and for all convex functions f which satisfy an additional technical condition. This technical condition is satisfied by the logarithmic function, and this fact can be used to deduce a sum-product estimate \begin{equation*}\max \{ |16A|, |A^{(16)}| \} \gg |A|^{\frac{3}{2} + c},\end{equation*} for some $c\gt 0$ . Previously, no sum-product estimate over $\mathbb R$ with exponent strictly greater than $3/2$ was known for any number of variables. Moreover, the technical condition on f seems to be satisfied for most interesting cases, and we give some further applications. In particular, we show that \begin{equation*}|AA| \leq K|A| \implies \,\forall d \in \mathbb R \setminus \{0 \}, \,\, |\{(a,b) \in A \times A : a-b=d \}| \ll K^C |A|^{\frac{2}{3}-c^{\prime}},\end{equation*} where $c,C \gt 0$ are absolute constants.