Malaysian Journal of Mathematical Sciences, January 2015, Vol. 9, No. 1
A Functional Equation with Conjugate Means Derived from a Weighted Arithmetic Mean
Annop Sonubon and Somsak Orankitjaroen
Corresponding Email: somsak.ora@mahidol.ac.th
Received date: -
Accepted date: -
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 $\varphi $ and $\psi$ on an open interval $I$ which is a solution of
$$\varphi^{-1}\left ( p\varphi (x) + q\varphi (y) + (1-p-q)\varphi (tx+(1-t)y)\right ) + \psi^{-1}\left ( r\psi (x) + s\psi (y) + (1-r-s)\psi (tx+(1-t)y)\right ) = x + y,$$
for all $x,y \in I$ where $p,q,r,s,t \in (0,1)$, $p \neq q$, $r \neq s$, $p+q \neq 1$, $r+s \neq 1$, $st = r(1-t)$ with either the conditions $p+q = r+s$ or $p+q = 2(r+s)$. We found that the solutions $\varphi $ and $\psi$ are in the form of linear functions.
Keywords: Mean, functional equation, conjugate mean
Indexing
