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In this paper, we prove that the bound \[ \max \{ |8A-7A|,|5f(A)-4f(A)| \}\gg |A|^{\frac{3}{2} + \frac{1}{54}-o(1)} \] holds for all $A \subset \mathbbR$, and for all convex functions $f$ which satisfy an additional technicalcondition. This technical condition is satisfied by the logarithmic function,and this fact can be used to deduce a sum-product estimate \[ \max \{ |16A| ,|A^{(16)}| \} \gg |A|^{\frac{3}{2} + c}, \] for some $c>0$. Previously, nosum-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 furtherapplications. In particular, we show that \[ |AA| \leq K|A| \implies \,\foralld \in \mathbb R \setminus \{0 \}, \,\, |\{(a,b) \in A \times A : a-b=d \}| \llK^C |A|^{\frac{2}{3}-c'}, \] where $c,C>0$ are absolute constants.

A better than $3/2$ exponent for iterated sums and products over $\mathbb R$