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

• Autorzy: Peter Kahlig , Janusz Matkowski
• Języki publikacji: EN

Abstrakty

EN

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. (original abstract)

Słowa kluczowe

EN

Mathematics; Mathematical functions; Convergence; Functional equation

PL

Matematyka; Funkcje matematyczne; Konwergencja; Równanie funkcyjne

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2020
• Tom: 34 (1)
• Strony: 104--122

Twórcy

autor

Peter Kahlig

• University of Vienna, Austria

autor

Janusz Matkowski

• University of Zielona Gora, Poland

Bibliografia

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• P. Kahlig and J. Matkowski, On the composition of homogeneous quasi-arithmetic means, J. Math. Anal. Appl. 216 (1997), 69-85.
• M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, Państwowe Wydawnictwo Naukowe and Uniwersytet Śląski, Warszawa-Kraków-Katowice, 1985; Second edition, edited and with a preface by A. Gilányi, Birkhäuser Verlag, Basel, 2009.
• J. Matkowski, Invariant and complementary quasi-arithmetic means, Aequationes Math. 57 (1999), 87-107.
• J. Matkowski, Iterations of mean-type mappings and invariant means, Ann. Math. Sil. 13 (1999), 211-226.
• J. Matkowski, Iterations of the mean-type mappings, in: A.N. Sharkovsky and I.M. Sushko (eds.), Iteration Theory (ECIT'08), Grazer Mathematische Berichte, 354, Karl-Franzens-Universität Graz, Graz, 2009, pp. 158-179.
• J. Matkowski, Generalized weighted arithmetic means, in: Th.M. Rassias and J. Brzdęk (eds.), Functional Equations in Mathematical Analysis, Springer, New York, 2012, pp. 563-582.
• J.S. Ume and Y.H. Kim, Some mean values related to the quasi-arithmetic mean, J. Math. Anal. Appl. 252 (2000), 167-176.

Identyfikatory

DOI

10.2478/amsil-2020-0015