Isomorphic pairs of homogeneous functions and their morphisms

Summary. In this paper we determine all iseomorphic pairs (isomorphicpairs with monotonic, thus continuous isomorphisms) ofcontinuous, strictly increasing, linearly homogeneous functions defined oncartesian squares I ² and J ² of intervals of positive numbers or on their restrictions or and or We prove that, if the iseomorphy is nontrivial, then eachhomogeneous function is a (weighted) geometric or power mean or ajoint pair of such means.In functional equations terminology this means that all nontrivialcontinuous strictly increasing linearly homogeneous solutions G, H(with the continuous strictly monotonic F also unknown) of theequation on D _< or D _>are weighted geometric or power means, while onI ² they are joint pairs of weighted geometric means or of weightedpower means.