A note on equivalence of means

LUCIO R. BERRONE; ARIEL L. LOMBARDI

PUBLICATIONES MATHEMATICAE-DEBRECEN

University of Debrecen

Lugar: Debrecen; Año: 2001 vol. 58 p. 49 - 56

0033-3883

Given a real interval $I$, a relation, denoted by `$\sim$', is defined on the set of means on $I\times I$ by setting $M\sim N$ when there exists a surjective continuous function $f$ solving the functional equation $$f(M(x,y))=N(f(x),f(y)),\quad x,y\in I.$$ A surjective and continuous solution to this equation turns out to be injective and so, `$\sim$' is an equivalence. This fact seems to be not properly noticed in the literature on means.