Invariant and complementary quasi-arithmetic means
If \(I\) is a proper (non-singleton) real interval and \(M\) and \(N\) are continuous, both map \(I^{2}\) into \(I\), and both \(M(x,y)\) and \(N(x,y)\) lie between \(\min(x,y)\) and \(\max(x,y),\) one of them always strictly between if \(x\neq y\) (that is, both are means and one of them is a strict mean), then it is easy to see that there exists a unique mean \(K\) such that \(K[M(x,y),N(x,y)]=K(x,y)\) for all \(x,y\) in \(I.\) The problem is more difficult if \(M,N,K\) are all supposed to be quasi-arithmetic, that is, of the form \(f^{-1}([f(x)+f(y)]/2)\) (the continuous bijection \(f\) is the generating function). The author offers the general forms of generating functions of \(M\) and \(N\) if one of them is twice continuously differentiable.
scientific article; zbMATH DE number 1295532