A generalized a-Wright convexity and related functional equation
Abstract
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.
Keywords
a-Wright convexity; Jensen convexity; semicontinuity; functional equation; functional inequality
References
2. Gy. Maksa, K. Nikodem, Zs. Páles, Results on t-Wright convexity, C. R. Math. Rep. Acad. Sci. Canada, Vol. 13 (1991), 274-278.
3. J. Matkowski, On a-Wright convexity and the converse of Minkowski's inequality, Aequationes Math. 43 (1992), 219-224.
4. Zs. Páles, On two variable functional inequalities, C. R. Math. Rep. Acad. Sci. Canada, Vol. 10 (1988), 25-28.
Katedra Matematyki, Filia Politechniki Łódzkiej w Bielsku-Białej Poland
Instytut Matematyki, Wyższa Szkoła Pedagogiczna w Częstochowie Poland
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