  In this paper we prove new bounds for sums of convex or concave functions.
Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and
for all $f,g$ convex or concave functions, we have
  $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$
  This result can be used to obtain bounds on a number of two-variable
expanders of interest, as well as to the asymmetric sum-product problem. We
also adjust our technique to also prove the three-variable expansion result
  \[
  |AB+A|\gtrsim |A|^{\frac32 +\frac3{170}}\,.
  \]
  Our methods follow a series of recent developments in the sum-product
literature, presenting a unified picture. Of particular interest is an
adaptation of a regularisation technique of Xue, that enables us to find
positive proportion subsets with certain desirable properties.
