Quotient mean, its invariance with respect to a quasi-arithmetic mean-type mapping, and some applications (Q657822)

The author introduces the so-called quotient mean \[ Q_{f,g}(x,y):=\left( \frac{f}{g} \right) ^{-1} \left( \frac{f(x)} {g(y)} \right) \] for functions \(f\) and \(g\) defined on an interval and satisfying certain conditions. The classical arithmetic, geometric and harmonic means are particular cases. The invariance of this mean with respect to the weighted quasi-arithmetic mean is examined. Applications to the iterations of means are pointed out, too.