A better than exponent for iterated sums and products over

Abstract In this paper, we prove that the bound equation* \ |8A-7A|, |5f (A) -4f (A) | \ |A|^3{2 + 154}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* \ |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 \| KC |A|^2{3-c^}, equation* where c, C 0 are absolute constants.