Interval-type theorems concerning means

Paweł Pasteczka

Abstract


Each family M of means has a natural, partial order (point-wise order), that is M ≤ N iff M(x) ≤ N(x) for all admissible x.  
In this setting we can introduce the notion of interval-type set (a subset I ⊂ M such that whenever M ≤  P ≤ N for some M,N ∈ I and P ∈ M then P ∈ I). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. 
In the present paper we consider this property for Gini means and Hardy means. Moreover some results concerning L∞ metric among (abstract) means will be obtained.

Keywords


means; squeeze theorem; sandwich theorems; distance between means; Hardy means

Mathematics Subject Classification (2010)


26E60; 26D15

References


Bullen, Peter S. Handbook of means and their inequalities. Vol 560 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group, 2003.

Carleman, Torsten. "Sur les fonctions quasi-analitiques." Conférences faites au cinquième congrès des mathematiciens Scandinaves: tenu a Helsingfors du 4 au 7 juillet 1922, 181–196. Helsinki: Libr. Académique, 1932.

Daróczy, Zoltán and László Losonczi. "Über den Vergleich von Mittelwerten." Publ. Math. Debrecen 17 (1970): 289–297 (1971).

Duncan, John and Colin M. McGregor. "Carleman’s inequality." Amer. Math. Monthly 110, no. 5 (2003): 424–431.

Gini, Corrado. "Di una formula compressiva delle medie." Metron 13 (1938): 3–22.

Hardy, Godfrey H. "Note on a theorem of Hilbert." Math. Z. 6, no. 3-4 (1920): 314–317.

Hardy, Godfrey H. "Note on a theorem of Hilbert concerning series of positive terms." Proc. London Math. Soc. 23, no. 2 1925, Records of Proc. XLV–XLVI.

Knopp, Konrad. "Uber Reihen Mit Positiven Gliedern." J. London Math. Soc. 3, no. 3 (1928): 205–211.

Kufner, Alois, and Lech Maligranda, and Lars-Erik Persson. The Hardy Inequality: About Its History and Some Related Results. Vydavatelský servis, 2007.

Landau, Edmund. "A note on a theorem concerning series of positive terms." J. London Math. Soc. (1921): 138–39.

Pecaric, Josip, and Kenneth B. Stolarsky. "Carleman’s inequality: history and new generalizations." Aequationes Math. 61, no. 1-2 (2001): 49–62.

Páles, Zsolt. "Inequalities for sums of powers." J. Math. Anal. Appl. 131, no. 1 (1988): 265–270.

Páles, Zsolt. "Comparison of the Geometric Mean with Gini Means." in: "Report of Meeting - 15th International Conference on Functional Equations and nequalities." Ann. Univ. Paedagog. Crac. Stud. Math. 12 (2013): 91-131.

Páles, Zsolt, and Paweł Pasteczka. "Characterization of the Hardy property of means and the best Hardy constants." Math. Inequal. Appl. 19, no. 4, (2016): 1141–1158.

Páles, Zsolt, and Lars-Erik Persson. "Hardy-type inequalities for means." Bull. Austral. Math. Soc. 70, no. 3 (2004): 521–528.

Shisha, Oved, and Gerald T. Cargo. "On comparable means." Pacific J. Math. 14 (1964): 1053–1058.

Shisha, Oved, and Gerald T. Cargo. "A metric space connected with generalized means." J. Approximation Theory 2 (1969): 207–222.


Full Text: PDF

Download statistics: 1022

Licencja Creative Commons
This article by Paweł Pasteczka is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej