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.
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Vol. 10 (1996)
Published: 1996-09-30
10.1515/amsil
English