The Aumann functional equation for general weighting procedures

Abstract

The functional equation of composite type M(M(x; M(x; y)); M(M(x; y); y)) = M(x; y) arose in the course of the studies on the problem of extension and restriction of the number of arguments of a mean M performed by G. Aumann at the third decade of the past century. A solution to (1) in the analytic case was ulteriorly obtained by Aumann himself and remained as a noteworthy characterization of analytic quasiarithmetic means. An ample generalization of equation (1) which involves general weighting operators is considered in this paper. Under mild conditions on the regularity of the involved means, the general solution to this generalized equation is obtained for a particularly tractable class of weighting operators.