Abstract. Let $M$ be a mean on $[a,b]$ and let $\hat M(x,y):=x+y-M(x,y)$ $(x,y\in[a,b])$ be the mean which is complementary to $M$ with respect to the arithmetic mean. A function $f\colon[a,b]\to{\msbm R}$ is called {\it $M$-associate } if it possesses the following property: If $x,y\in[a,b]$ satisfy $M(x,y)=(x+y)/2$ and $f(x )=f\left((x+y)/2\right )$, then $f(y)=f(x)$. We consider the functional equation $$ f(M(x,y))=f(\hat M( x,y)) (x,y\in[a,b]) $$ with and without $f$ being $M$-associate.

AMS Subject Classification (1991): 39B22, 39B12, 26A18

Keyword(s): quasi-arithmetic mean, functional equation

Received May 3, 2000. (Registered under 2754/2009.)