Computer aided solution of the invariance equation for two-variable Gini means

One of the classical results of the regularity theory of convex functions is the theorem of F. Bernstein and G. Doetsch [1] which states that if a real valued Jensen-convex function defined on an open interval I is bounded from above on a subinterval of I then it is continuous. According to a related result by W. Sierpiński [3], the Lebesgue measurability of a Jensen-convex function implies its continuity, too. In this talk we generalize the theorems above for (M, N)-convex functions, calling a function f : I → J (M, N)-convex if it satisfies the inequality f(M(x, y)) ≤ Nx,y(f(x), f(y)) for all x, y ∈ I, where I and J are open intervals, M is a mean on I and Nx,y is a suitable mean on J for every x, y ∈ I (c.f., e.g., [2])