Computer assisted solution of an equality problem of mean values

Given weighted quasi-arithmetic means M, N the problem is when the mean of the formφ-1φ(x)+φ(y)-12(φ(M(x,y))+φ(N(x,y)))is again quasi-arithmetic. It was answered in [5] by Daróczy and Páles under the assumption that M=N is arithmetic. We investigate the complementary situation where at least one of the weighted arithmetic means M, N is not arithmetic and solve the problem locally, assuming that the generators of means are four times continuously differentiable. For the computation we use the software package Mathematica 4.0.