# Invariant and complementary quasi-arithmetic means

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## 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 \( M_{\gamma } \).

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