  In this paper we further study the relationship between convexity and
additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get
some improvements to earlier results of Elekes, Nathanson and Ruzsa
(\cite{ENR}). In particular, we show that for any finite set
$A\subset{\mathbb{R}}$ and any strictly convex or concave function $f$,
\[|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}\] and \[\max\{|A-A|,\
|f(A)+f(A)|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}.\] For the latter of
these inequalities, we go on to consider the consequences for a sum-product
type problem.
