A generalized a-Wright convexity and related functional equation - Matkowski, Janusz; Wróbel, Małgorzata (Annales Mathematicae Silesianae, 1996)
Let I be an interval and M,N: I×I→I some means with the strict internality property. Suppose that ϕ: I→ℝ is a non-constant and continuous solution of the functional equation ϕ(M(x,y)) + ϕ(N(x,y)) = ϕ(x) + ϕ(y). Then ϕ is one-to-one; moreover for every lower semicontinuous function f: I→ℝ satisfying the inequality f(M(x,y)) + f(N(x,y)) ≤ f(x) + f(y), the function f◦ϕ-1 is convex on ϕ(I). This is a generalization of an earlier result of Zs. Páles. An application to the a-Wright convex function is given.