Invariant and complementary quasi-arithmetic means

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 $M_{\gamma }$.