Mean invariance identity

For a continuous and increasing function $f$ in a real interval $I$, and a bivariable mean $P$ defined in $I^{2}$, we prescribe a pair of bivariable means $M$ and $N$ such that the quasiarithmetic mean $A_{f}$ generated by $f$ is invariant with respect to the mean-type mapping $\left(M,N\right)$. This allows to find effectively the limit of the iterates of the mean-type mapping $\left(M,N\right)$. The means $M$ and $N$ are equal iff $P$ is the arithmetic mean $A$; they are symmetric iff so so is $P$. Treating $f$ and $P$ as the parameters, we obtain the family of all pairs of means $\left(M,N\right)$ such that the quasiarithmetic mean $A_{f}$ is invariant with respect to $\left(M,N\right)$. In particular, we indicate the function $f$ and the mean $P$ such that the invariance identity $A_{f}\circ\left(M,N\right)=A_{f}$ coincides with the equality $G\circ\left(H,A\right)$, where $G$ and $H$ are the geometric and harmonic means, equivalent to the classical Pythagorean harmony proportion. Some examples and an application are also presented.