Measurability of (M, N)-Wright convex functions

Let I subset of R be an open interval and M, N : I(2) -> I be means on I. Let phi:I -> R be solution of the functional equation phi(M (x, y)) + phi(N (x, y)) = phi(x) + phi(y), x, y is an element of I. We give sufficient conditions on M, N and the function phi such that for every Lebesgue measurable solution f : I -> R of the functional inequality f(M(x, y)) + f(N(x, y)) <= f(x) + f(y), x, y is an element of I, the function f o phi(-1) : phi(I) -> R is convex.