Aequationes mathematicae

, Volume 74, Issue 3, pp 262–281 | Cite as

Homogeneous symmetric means of two variables



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.

Mathematics Subject Classification (2000).

Primary 39B22 


Mean value functional equation homogeneous function 


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Copyright information

© Birkhäuser Verlag AG 2007

Authors and Affiliations

  1. 1.Institute of MathematicsDebrecen UniversityDebrecenHungary

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