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

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.