Matkowski-Sutô type problem for conjugate arithmetic means

Abstract

Let I ⊂ R be an open interval and let CM(I) denote the set of all continuous and strictly monotonie real functions on I. For ϕ ϵ CM(I) we define A[*][ϕ](x,y) := ϕ[-1](ϕ(x) + ϕ(y) - ϕ(x+y/2) for all x,y ϵ I, which is called a conjugate arithmetic mean on I. In this paper we solve the functional equation A[*][ϕ](x,y) + A[*][ϕ](x,y) = x + y for all x,y ε I, where ϕ,ψ ϵ CM [I) and either ϕ or ψ is twice continuously differentiable on I.