Explicit forms of invariant means: complementary results to Gauss left( {mathcal {A}},{mathcal {G}}right) -theorem and some applications

Abstract

Explicit forms of invariant means for all mean-type mappings generated by the bivariable classical arithmetic, geometric and harmonic means (complementary to the Gauss mathcal {AGM}-theorem) are given. A generalization for higher dimension mean-type mappings as well as two open problems and an application in determining implicit solutions of some functional equations, are presented.