On means generated through the Cauchy mean value theorem (Q1587831)

The authors offer several results about Cauchy means \(\mu\) defined by \(F'(\mu(x,y))/G'(\mu(x,y)) = (F(y)-F(x))/(G(y)-G(x))\) and their relation to quasiarithmetic means \(f^{-1}((f(x)+f(y))/2).\) In particular, every Cauchy mean is the fixed point of a mixture of two quasiarithmetic means \(M,N,\) where the ``mixing operator'' is, roughly speaking, given by \[ M(\mu[N(x,y),y], \mu[x,N(x,y)]). \] Particular cases of Cauchy means are the means of \textit{E. B. Leach} and \textit{M. C. Sholander} [Am. Math. Mon. 85, 84-90 (1978; Zbl 0379.26012); J. Math. Anal. Appl. 92, 207-223 (1983; Zbl 0517.26007)] and the quasiarithmetic means themselves.