A better than exponent for iterated sums and products over

In this paper, we prove that the bound {equation*}max \{ |8A-7A|,|5f(A)-4f(A)| \} |A|3/2 + 1/54{equation*} holds for all A ⊂ 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 {equation*}max \{ |16A|, |A⁽¹⁶⁾| \} |A|3/2 + c,{equation*} for some c> 0 . Previously, no sum-product estimate over 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 {equation*}|AA| ≤ K|A| \,∀ d ∈ R \{0 \}, \,\, |\{(a,b) ∈ A × A : a-b=d \}| K^C |A|^{2/3-c^{}},{equation*} where c,C > 0 are absolute constants.