Generalization of the harmonic weighted mean via Pythagorean invariance identity and application

Peter Kahlig; Janusz Matkowski

Published: 2020-07-09

Vol. 34 No. 1 (2020)

Generalization of the harmonic weighted mean via Pythagorean invariance identity and application

Peter Kahlig , Janusz Matkowski

Abstract

Under some simple conditions on the real functions f and g defined on an interval I⊂(0,∞), the two-place functions A_f(x; y) = f (x)+y-f (y) and Gg(x; y) = \frac{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 H_f,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of H_f,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.

Keywords:

generalized arithmetic and geometric means , invariance identity , generalized harmonic mean , functional equations , mean-type mappings , iteration , convergence of iterates , invariant functions

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Kahlig, P., & Matkowski, J. (2020). Generalization of the harmonic weighted mean via Pythagorean invariance identity and application. Annales Mathematicae Silesianae, 34(1), 104–122. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13636

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Peter Kahlig

Affiliation:

Department of Meteorology and Geophysics, University of Vienna, Austria Austria

Janusz Matkowski

j.matkowski@wmie.uz.zgora.pl
https://orcid.org/0000-0003-0011-0579

Affiliation:

Wydział Matematyki, Informatyki i Ekonometrii, Uniwersytet Zielonogórski Poland

Vol. 34 No. 1 (2020)
Published: 2020-07-20

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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