aequationes mathematicae

, Volume 57, Issue 1, pp 87–107

# Invariant and complementary quasi-arithmetic means

• J. Matkowski

## Summary.

Let $$I\subset {\Bbb R}$$ be an interval. Under the assumptions that $$\phi, \psi, \gamma :I \rightarrow {\Bbb R}$$ are one-to-one, $$\gamma(I)$$ is an interval, and at least one of the functions $$\phi \circ \gamma^{-1}$$, $$\psi \circ \gamma ^{-1}$$ is twice continuously differentiable on $$\gamma(I)$$, we determine all the quasi-arithmetic means $$M_{\gamma}$$, $$M_{\phi }$$, $$M_{\psi }$$ satisfying the functional equation $$M_{\gamma}(M_{\phi }(x,y)$$, $$M_{\psi }(x,y)) = M_{\gamma}(x,y)$$ which can be interpreted in the following two ways: the mean $$M_{\gamma }$$ is invariant with respect to $$M_{\phi }$$ and $$M_{\psi }$$, or $$M_{\psi }$$ is ‘complementary’ to $$M_{\phi }$$ with respect to the mean $$M_{\gamma }$$.

Keywords. Means, invariant mean, complementary means with respect to a given mean, quasiarithmetic means, positively homogeneous mean, composition of means, functional equations.