Interval-type theorems concerning means

Pasteczka, Paweł

doi:10.2478/aupcsm-2018-0004

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.

References

[1] Bullen, Peter S. Handbook of means and their inequalities. Vol 560 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group, 2003. Cited on 38.
Search in Google Scholar Back to article
[2] 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. Cited on 41.
Search in Google Scholar Back to article
[3] Daróczy, Zoltán and László Losonczi. “Über den Vergleich von Mittelwerten.” Publ. Math. Debrecen 17 (1970): 289–297 (1971). Cited on 40.10.5486/PMD.1970.17.1-4.33
Search in Google Scholar Back to article
[4] Duncan, John and Colin M. McGregor. “Carleman’s inequality.” Amer. Math. Monthly 110, no. 5 (2003): 424–431. Cited on 41.10.1080/00029890.2003.11919980
Search in Google Scholar Back to article
[5] Gini, Corrado. “Di una formula compressiva delle medie.” Metron 13 (1938): 3–22. Cited on 40.
Search in Google Scholar Back to article
[6] Hardy, Godfrey H. “Note on a theorem of Hilbert.” Math. Z. 6, no. 3-4 (1920): 314–317. Cited on 41.10.1007/BF01199965
Search in Google Scholar Back to article
[7] 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. Cited on 41.
Search in Google Scholar Back to article
[8] Knopp, Konrad. “Uber Reihen Mit Positiven Gliedern.” J. London Math. Soc. 3, no. 3 (1928): 205–211. Cited on 41.10.1112/jlms/s1-3.3.205
Search in Google Scholar Back to article
[9] Kufner, Alois, and Lech Maligranda, and Lars-Erik Persson. The Hardy Inequality: About Its History and Some Related Results. Vydavatelský servis, 2007. Cited on 41.
Search in Google Scholar Back to article
[10] Landau, Edmund. “A note on a theorem concerning series of positive terms.” J. London Math. Soc. (1921): 138–39. Cited on 41.
Search in Google Scholar Back to article
[11] Pečarić, Josip, and Kenneth B. Stolarsky. “Carleman’s inequality: history and new generalizations.” Aequationes Math. 61, no. 1-2 (2001): 49–62. Cited on 41.10.1007/s000100050160
Search in Google Scholar Back to article
[12] Páles, Zsolt. “Inequalities for sums of powers.” J. Math. Anal. Appl. 131, no. 1 (1988): 265–270. Cited on 39.10.1016/0022-247X(88)90204-1
Search in Google Scholar Back to article
[13] Páles, Zsolt. “Comparison of the Geometric Mean with Gini Means.” in: “Report of Meeting - 15th International Conference on Functional Equations and Inequalities.” Ann. Univ. Paedagog. Crac. Stud. Math. 12 (2013): 91-131. Cited on 39.
Search in Google Scholar Back to article
[14] 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. Cited on 41.10.7153/mia-19-84
Search in Google Scholar Back to article
[15] Páles, Zsolt, and Lars-Erik Persson. “Hardy-type inequalities for means.” Bull. Austral. Math. Soc. 70, no. 3 (2004): 521–528. Cited on 41.10.1017/S0004972700034778
Search in Google Scholar Back to article
[16] Shisha, Oved, and Gerald T. Cargo. “On comparable means.” Pacific J. Math. 14 (1964): 1053–1058. Cited on 42.10.2140/pjm.1964.14.1053
Search in Google Scholar Back to article
[17] Shisha, Oved, and Gerald T. Cargo. “A metric space connected with generalized means.” J. Approximation Theory 2 (1969): 207–222. Cited on 42.10.1016/0021-9045(69)90041-0
Search in Google Scholar Back to article

Interval-type theorems concerning means

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