Homogeneous symmetric means of two variables

Summary.

Let $f, g: I \rightarrow{\mathbb{R}}$ be given continuous functions on the interval I such that g ≠ 0, and $h :=\frac{f}{g}$ is strictly monotonic (thus invertible) on I. Taking an increasing nonconstant function μ on [0, 1]

$$ M_{f,g,\mu}(x, y) := h^{-1}\left(\frac{\int \limits_0^1f(tx + (1-t)y)\,d\mu(t)}{\int \limits_0^1g(tx + (1- t)y)\,d\mu(t)}\right) (x, y \in I)$$

is a mean value of $x, y \in I$. Here we solve the homogeneity equation

$$M_{f,g,\mu}(tx, ty) = tM_{f,g,\mu}(x, y)\quad (x, y \in I, t \in I_x\cap I_y)$$

for two important special cases of symmetric means of this type: for the quasi-arithmetic means weighted by a weight function and for the Cauchy means. We assume that $I\subset ]0,\infty[$ is open, $1\in I$, and f, g satisfy strong differentiability conditions.