Aequationes mathematicae

, Volume 74, Issue 3, pp 262–281

# Homogeneous symmetric means of two variables

• László Losonczi
Article

## 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.

Primary 39B22

## Keywords.

Mean value functional equation homogeneous function