A real mean is a symmetric function $M : \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ with $M(x,x)=x$, which is monotonuos in each variable.
According to Wikipedia: https://en.wikipedia.org/wiki/Quasi-arithmetic_mean, and the german paper "Aufbau von Mittelwerten mehrerer Argumente. II." in which Georg Augmann proved in 1935 that for a mean that is an analytic and symmetric function, the balancing property $$ M(M(x,M(x,y)),M(y,M(x,y)))=M(x,y) $$ implies, that $M$ is a quasi-arithmetic mean, thus $$ M(x,y)=f^{-1}(\frac{f(x)+f(y)}{2}) $$ for an monotonous, continous function $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$.
For a general weighted quasi-arithmetic mean: $$ M(x,y)=f^{-1}(w\cdot f(x)+(1-w)\cdot f(y)), $$ the balancing property is obviously not always true. Instead one can show rather easily, that some form of medialty property is true: $$ M(M(a,b),M(c,d))= M(M(a,c),M(b,d)). $$
If one assumes the medialty property (for $a<b<c<d$ or in general) are there any results like Augmann's? Is it possible to show that a mean which fulfills this condition and an analytic function is a weighted, quasi-artihmetic mean? Are there any related conditions impylying this form?
$Edit:$ a quasi-arithmetic mean with $w\neq 0.5$ obviously doesn't satisfy symmetry any more and is thus no mean according to my previous unthoughtful definition.
Theorem 1 on page 571 in this paper seems to answer my question. I might elaborate, once i figured out how to read their reported source and understand it.