A functional equation with conjugate means derived from a weighted arithmetic mean

Abstract

In this paper, we seek a solution of a functional equation with conjugate means derived from a weighted arithmetic mean; that is, finding continuous strictly monotonic functions π{variant} and ψ on an open interval I which is a solution of π{variant}^-1(pπ{variant}(x)+qπ{variant}(y)+(1-p-q)π{variant}(tx+(1-t)y)) +ψ^-1(rψ(x)+sψ(y)+(1-r-s)ψ(tx+(1-t)y))=x+y, for all x,y ε{lunate} 1 where p,q,r,s,tε{lunate}(0,1),p≠q, r≠s, p+q≠1, r+s≠1, st=r(1-t) with either the conditions p+q=r+s or p+q=2(r+s). We found that the solutions π{variant} and ψ are in the form of linear functions.