be given continuous functions such that g(x)≠ 0 for x ∈I and h(x) : = f(x)/g(x) (x ∈I) is strictly monotonic (thus invertible) on I. Let further μ be an increasing non-constant function on [0, 1]. We consider the function
i.e. Mf, g, μ is a mean on I. With suitable choice of μ we can get from it the quasi-arithmetic mean weighted by a weight function, and also the Cauchy or difference mean.
The equality problem for these two classes of means has been solved in , . The aim of this paper is twofold. First, we solve these equality problems in a unified way, second we get rid of the inconvenient conditions (vanishing or not vanishing of some functions) posed in the previous papers.
Mathematics Subject Classification (2000).
Divided differences mean value functional equation