A Note on the Sum-Product Problem and the Convex Sumset Problem

We provide a new exponent for the Sum-Product conjecture on $\mathbb{R}$. Namely for $A\subset \mathbb{R}$ finite, $$\max \{\mid A+A\mid ,\mid AA\mid \}{\gg }_{\epsilon }{\mid A\mid }^{\frac{4}{3}+\frac{10}{4407}-\epsilon }\mathrm{.}$$ We also provide new exponents for $A\subset \mathbb{R}$ finite and convex, namely $$\mid A+A\mid {\gg }_{\epsilon }{\mid A\mid }^{\frac{46}{29}-\epsilon },$$ and $$\mid A-A\mid {\gg }_{\epsilon }{\mid A\mid }^{\frac{8}{5}+\frac{1}{3440}-\epsilon }\mathrm{.}$$