On iteration semigroups of mean-type mappings and invariant means (Q1861183)

This is in many respects continuation of the author's paper [Ann. Math. Sil. 13, 211-226 (1999; Zbl 0954.26015)], and familiarity with the previous paper helps understanding the present one. The following is offered as one of the two main results. Let \(M^t(x,y):= (M_1^t(x,y),M_2^t(x,y))\) be a pair of continuous strict mean values (\(\min(x,y)<M_j^t(x,y)<\max(x,y)\) \((j=1,2)\) if \(x\neq y\)) on the Cartesian square of a real interval \(I\), continuous also in the positive parameter \(t\). If (o) \(M^s\circ M^t =M^{s+t}\) holds for the composition \(\circ\) then there exists a unique continuous, necessarily strict mean \(K\) such that \(K\circ M^t=K\) for all \(t>0\). The other main result examines when (o) holds for quasiarithmetic weighted means \(M_j^t(x,y)=f^{-1}[a_j^t f(x)+(1-a_j^t) f(y)]\) (\(a_j^t\in]0,1[\); \(j=1,2\); \(t>0\); \(x\in I\), \(y\in I)\).