<abstract xmlns="http://www.w3.org/1999/xhtml">Under some simple conditions on the real functions <italic>f</italic> and <italic>g</italic> defined on an interval <italic>I</italic> ⊂ (0, <italic>∞</italic>), the two-place functions <italic>Af</italic> (<italic>x, y</italic>) = <italic>f</italic> (<italic>x</italic>) + <italic>y − f</italic> (<italic>y</italic>) and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0015_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>G</mi><mi>g</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></mfrac><mi>y</mi></mrow></math><tex-math>{G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}y</tex-math></alternatives></inline-formula> generalize, respectively, <italic>A</italic> and <italic>G</italic>, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity <italic>G</italic> ∘ (<italic>H, A</italic>) = <italic>G</italic> (equivalent to the Pythagorean harmony proportion), a suitable weighted extension <italic>Hf,g</italic> of the classical harmonic mean <italic>H</italic> is introduced. An open problem concerning the symmetry of <italic>Hf,g</italic> is proposed. As an application a method of effective solving of some functional equations involving means is presented.</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)=g(x)g(y)y{G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}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.

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

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

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...

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

Published Online: Jul 09, 2020

Page range: 104 - 122

Received: Jul 20, 2019

Accepted: Jun 03, 2020

DOI: https://doi.org/10.2478/amsil-2020-0015

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

© 2020 Peter Kahlig et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Published Online: Jul 09, 2020

Page range: 104 - 122

Received: Jul 20, 2019

Accepted: Jun 03, 2020

DOI: https://doi.org/10.2478/amsil-2020-0015

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

© 2020 Peter Kahlig et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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