An Optimal Double Inequality for Means
© W.-M. Qian and N.-G. Zheng. 2010
Received: 3 September 2010
Accepted: 27 September 2010
Published: 30 November 2010
For , the generalized logarithmic mean , arithmetic mean and geometric mean of two positive numbers and are defined by , ; , , , ; , , ; , , ; and , respectively. In this paper, we give an answer to the open problem: for , what are the greatest value and the least value , such that the double inequality holds for all ?
It is wellknown that is continuous and increasing with respect to for fixed and . In the recent past, the generalized logarithmic mean has been the subject of intensive research. Many remarkable inequalities and monotonicity results can be found in the literature [1–9]. It might be surprising that the generalized logarithmic mean, has applications in physics, economics, and even in meteorology [10–13].
The proof of the following Theorem B can be found in .
The following Theorems C–E were established by Alzer and Qiu in .
In order to establish our main result, we need two lemmas, which we present in this section.
Therefore, Lemma 2.1 follows from (2.3)–(2.6) and (2.8).
3. Main Results
This work was supported by the Natural Science Foundation of Zhejiang Broadcast and TV University (Grant no. XKT-09G21).
- Stolarsky KB: The power and generalized logarithmic means. The American Mathematical Monthly 1980, 87(7):545–548. 10.2307/2321420MathSciNetView ArticleMATHGoogle Scholar
- Pearce CEM, Pečarić J: Some theorems of Jensen type for generalized logarithmic means. Revue Roumaine de Mathématiques Pures et Appliquées 1995, 40(9–10):789–795.MathSciNetMATHGoogle Scholar
- Mond B, Pearce CEM, Pečarić J: The logarithmic mean is a mean. Mathematical Communications 1997, 2(1):35–39.MathSciNetMATHGoogle Scholar
- Chen Ch-P, Qi F: Monotonicity properties for generalized logarithmic means. The Australian Journal of Mathematical Analysis and Applications 2004, 1(2, article 2):1–4.MathSciNetMATHGoogle Scholar
- Xia W-F, Chu Y-M, Wang G-D: The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means. Abstract and Applied Analysis 2010, 2010:-9.Google Scholar
- Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10.Google Scholar
- Chu Y-M, Xia W-F: Inequalities for generalized logarithmic means. Journal of Inequalities and Applications 2009, 2009:-7.Google Scholar
- Chen Ch-P: The monotonicity of the ratio between generalized logarithmic means. Journal of Mathematical Analysis and Applications 2008, 345(1):86–89. 10.1016/j.jmaa.2008.03.071MathSciNetView ArticleMATHGoogle Scholar
- Qi F, Li X-A, Chen S-X: Refinements, extensions and generalizations of the second Kershaw's double inequality. Mathematical Inequalities & Applications 2008, 11(3):457–465.MathSciNetView ArticleMATHGoogle Scholar
- Kahlig P, Matkowski J: Functional equations involving the logarithmic mean. Zeitschrift für Angewandte Mathematik und Mechanik 1996, 76(7):385–390. 10.1002/zamm.19960760710MathSciNetView ArticleMATHGoogle Scholar
- Pittenger AO: The logarithmic mean in variables. The American Mathematical Monthly 1985, 92(2):99–104. 10.2307/2322637MathSciNetView ArticleMATHGoogle Scholar
- Nadirashvili NS: New isoperimetric inequalities in mathematical physics. In Partial Differential Equations of Elliptic Type (Cortona, 1992), Sympos. Math., XXXV. Cambridge University Press, Cambridge, UK; 1994:197–203.Google Scholar
- Pólya G, Szegö G: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton, NJ, USA; 1951:xvi+279.Google Scholar
- Alzer H, Janous W: Solution of problem . Crux Mathematicorum 1987, 13: 173–178.Google Scholar
- Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities, Mathematics and Its Applications (East European Series). Volume 31. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.Google Scholar
- Janous W: A note on generalized Heronian means. Mathematical Inequalities & Applications 2001, 4(3):369–375.MathSciNetView ArticleMATHGoogle Scholar
- Carlson BC: The logarithmic mean. The American Mathematical Monthly 1972, 79: 615–618. 10.2307/2317088MathSciNetView ArticleMATHGoogle Scholar
- Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983, 92(1):207–223. 10.1016/0022-247X(83)90280-9MathSciNetView ArticleMATHGoogle Scholar
- Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471–473. 10.1007/BF01200091MathSciNetView ArticleMATHGoogle Scholar
- Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986, 47(5):422–426. 10.1007/BF01189983MathSciNetView ArticleMATHGoogle Scholar
- Alzer H, Qiu S-L: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201–215. 10.1007/s00013-003-0456-2MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.