Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application

Peter Kahlig 1  and Janusz Matkowski 2
  • 1 Department of Meteorology and Geophysics, University of Vienna, A 1090, Vienna, Austria
  • 2 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516, Zielona Góra, Poland


Under some simple conditions on the real functions f and g defined on an interval I ⊂ (0, ), the two-place functions Af (x, y) = f (x) + y − f (y) and Gg(x,y)=g(x)g(y)y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ∘ (H, A) = G (equivalent to the Pythagorean harmony proportion), a suitable weighted extension Hf,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of Hf,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.

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