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

In this paper, we prove that the bound $$\max \{\mathrm{\mid }8A-7A\mathrm{\mid },\mathrm{\mid }5f(A)-4f(A)\mathrm{\mid }\}\gg \mathrm{\mid }A{\mathrm{\mid }}^{\frac{3}{2}+\frac{1}{54}-o(1)}$$ 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 $$\max \{\mathrm{\mid }16A\mathrm{\mid },\mathrm{\mid }{A}^{(16)}\mathrm{\mid }\}\gg \mathrm{\mid }A{\mathrm{\mid }}^{\frac{3}{2}+c},$$ for some $c>0$. Previously, no sum-product estimate over $\mathbb{R}$ with exponent strictly greater than $3\mathrm{/}2$ 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 $$\mathrm{\mid }AA\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }⟹\mathrm{\forall }d\in \mathbb{R}\setminus \{0\},\mathrm{\mid }\{(a,b)\in A\times A:a-b=d\}\mathrm{\mid }\ll K^C\mathrm{\mid }A{\mathrm{\mid }}^{\frac{2}{3}-c^{\mathrm{\prime }}},$$ where $c,C>0$ are absolute constants.