Isomorphic pairs of homogeneous functions and their morphisms

Summary.

In this paper we determine all iseomorphic pairs (isomorphic pairs with monotonic, thus continuous isomorphisms) of continuous, strictly increasing, linearly homogeneous functions defined on cartesian squares I ² and J ² of intervals of positive numbers or on their restrictions $D_{<} := \{(x, y) \in I^{2}\, \vert\: x \leq y\}$ or $D_{>} := \{(x, y) \in I^{2}\, \vert\: x \geq y\},$ and $\{(u, \upsilon) \in J^{2}\, \vert\: u \leq \upsilon\}$ or $\{(u, \upsilon) \in J^{2}\, \vert\: u \geq \upsilon\}.$ We prove that, if the iseomorphy is nontrivial, then each homogeneous function is a (weighted) geometric or power mean or a joint pair of such means.

In functional equations terminology this means that all nontrivial continuous strictly increasing linearly homogeneous solutions G, H (with the continuous strictly monotonic F also unknown) of the equation

$F[G(x, y] = H[F(x), F(y)]$

on D _< or D _> are weighted geometric or power means, while on I ² they are joint pairs of weighted geometric means or of weighted power means.