Generalization of the harmonic weighted mean via Pythagorean invariance identity and application
Abstract
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) = \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 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.
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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Department of Meteorology and Geophysics, University of Vienna, Austria Austria
Wydział Matematyki, Informatyki i Ekonometrii, Uniwersytet Zielonogórski Poland
https://orcid.org/0000-0003-0011-0579
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