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
AβR and any strictly convex or concave function f,
β£A+f(A)β£β«(logβ£Aβ£)2/19β£Aβ£24/19β and max{β£AβAβ£,Β β£f(A)+f(A)β£}β«(logβ£Aβ£)2/11β£Aβ£14/11β. For the latter of
these inequalities, we go on to consider the consequences for a sum-product
type problem