Invariance of means (Q1784262)

Let \(M\) and \(N\) be means on the same interval \(I\). The paper deals with the following invariance problem: finding a mean \(K\) on \(I\) such that \[ K(M(x,y),N(x,y))=K(x,y), \] for all \(x,y\in I\). One can see as a starting point of this problem the identity \[ \frac{x+y}{2}\cdot \frac{2}{\frac{1}{x}+\frac{1}{y}}=xy. \] In particular, the authors focus their attention on quasi-arithmetic means, Bajraktarević means and Cauchy means. The paper provides an interesting survey of the research on this topic and a comprehensive list of references.