On homogeneous quasideviation means (Q1114872)

The author [Acta Math. Acad. Sci. Hung. 40, 243-260 (1982; Zbl 0541.26006)] has generalized the concept of quasiarithmetic means with weight function introducing the concept of a quasideviation E on an interval I and the quasideviation mean \({\mathfrak M}_ E\). A function E: \(I\times I\to {\mathbb{R}}\) is called a quasideviation on \(I\subset R_+\) if \(sgn E(x,y)=sgn(x-y),\) for \(x,y\in I\), \(y\to E(x,y)\) is continuous in I for each \(x\in I\) and \(y\to E(x,y)/E(x',y)\) is strictly decreasng on (x,x') for \(x<x'\) in I. If \(x_ 1,...,x_ n\in I\), then the equation \(E(x_ 1,y)+...+E(x_ n,y)=0\) has a unique solution \(y=y_ 0\) which is between \(\min_{1\leq i\leq n}x_ i\) and \(\max_{1\leq i\leq n}x_ i\). This value is denoted by \({\mathfrak M}_ E(x_ 1,...,x_ n)\). In the paper the author describes the structure of quasideviations E that generate homogeneous means \({\mathfrak M}_ E\), i.e. such that \({\mathfrak M}_ E(tx_ 1,...,tx_ n)=t{\mathfrak M}_ E(x_ 1,...,x_ n)\) for all \(n\in {\mathbb{N}}\), \(x_ 1,...,x_ n\in I\) with \(tx_ 1,...,tx_ n\in I\).