- Research Article
- Open access
- Published:
An Optimal Double Inequality for Means
Journal of Inequalities and Applications volume 2010, Article number: 578310 (2010)
Abstract
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 ?
1. Introduction
For , the generalized logarithmic mean of two positive numbers and is defined by
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].
If we denote by ,, , and the arithmetic mean, identric mean, logarithmic mean, geometric mean and harmonic mean of two positive numbers and , respectively, then
For , the th power mean of two positive numbers and is defined by
In [14], Alzer and Janous established the following sharp double inequality (see also [15], Page 350):
for all .
For , Janous [16] found the greatest value and the least value such that
for all .
In [17–19] the authors present bounds for and in terms of and .
Theorem A.
For all positive real numbers and with , one has
The proof of the following Theorem B can be found in [20].
Theorem B.
For all positive real numbers and with , one has
The following Theorems C–E were established by Alzer and Qiu in [21].
Theorem C.
The inequalities
hold for all positive real numbers and with if and only if and .
Theorem D.
Let and be real numbers with . If , then
And if , then
Theorem E.
For all real numbers and with , one has
with the best possible parameter .
However, the following problem is still open: for , what are the greatest value and the least value , such that the double inequality
holds for all ? The purpose of this paper is to give the solution to this open problem.
2. Lemmas
In order to establish our main result, we need two lemmas, which we present in this section.
Lemma 2.1.
If , then
Proof.
Let , then simple computation yields
where
where
If , then from (2.7) we clearly see that
Therefore, Lemma 2.1 follows from (2.3)–(2.6) and (2.8).
Lemma 2.2.
If , then
Proof.
Let , then simple computation leads to
where
where
where
If , then from (2.14) we clearly see that
From (2.10)–(2.13) and (2.15) we know that for .
3. Main Results
Theorem 3.1.
If , then for all , with equality if and only if , and the constant in , cannot be improved.
Proof.
If , then we clearly see that .
If , without loss of generality, we assume that . Let and
Firstly, we prove . The proof is divided into three cases.
Case 1.
. We note that (1.1) leads to the following identity:
From (3.2) and Lemma 2.1 we clearly see that for and .
Case 2.
. Equation (1.1) leads to the following identity:
From (3.3) and Lemma 2.2 we clearly see that for and .
Case 3.
. From (1.1) we have the following identity:
Equation (3.4) and elementary computation yields
where
If , then (3.8) implies
for . Therefore, follows from (3.5)–(3.7) and (3.9).
If , then (3.8) leads to
for . Therefore, follows from (3.5)–(3.7) and (3.10).
Next, we prove that the constant in the inequality cannot be improved. The proof is divided into five cases.
Case 1.
. For any , let , then (1.1) leads to
where .
Making use of Taylor expansion we get
Case 2.
. For any , let , then
where .
Using Taylor expansion we have
Case 3.
. For any , let , then
where .
Making use of Taylor expansion and elaborated calculation we have
Case 4.
. For any , let , then
where .
Using Taylor expansion and elaborated calculation we have
Case 5.
. For any , let , then
where .
Using Taylor expansion and elaborated calculation we get
Cases 1–5 show that for any , there exists , for any there exists such that for .
Theorem 3.2.
If , then for all , with equality if and only if , and the constant in cannot be improved.
Proof.
If , then we clearly see that .
If , without loss of generality, we assume that . Let and
Firstly, we prove for . Simple computation leads to
where
for and .
From (3.23) we clearly see that
for .
Since , we have for . Therefore, follows from (3.22) and (3.24).
Next, we prove that the constant cannot be improved.
For any , we have
Equation (3.25) imply that for any there exists , such that for .
References
Stolarsky KB: The power and generalized logarithmic means. The American Mathematical Monthly 1980, 87(7):545–548. 10.2307/2321420
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.
Mond B, Pearce CEM, Pečarić J: The logarithmic mean is a mean. Mathematical Communications 1997, 2(1):35–39.
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.
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.
Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10.
Chu Y-M, Xia W-F: Inequalities for generalized logarithmic means. Journal of Inequalities and Applications 2009, 2009:-7.
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.2008.03.071
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.
Kahlig P, Matkowski J: Functional equations involving the logarithmic mean. Zeitschrift für Angewandte Mathematik und Mechanik 1996, 76(7):385–390. 10.1002/zamm.19960760710
Pittenger AO: The logarithmic mean in variables. The American Mathematical Monthly 1985, 92(2):99–104. 10.2307/2322637
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.
Pólya G, Szegö G: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton, NJ, USA; 1951:xvi+279.
Alzer H, Janous W: Solution of problem . Crux Mathematicorum 1987, 13: 173–178.
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.
Janous W: A note on generalized Heronian means. Mathematical Inequalities & Applications 2001, 4(3):369–375.
Carlson BC: The logarithmic mean. The American Mathematical Monthly 1972, 79: 615–618. 10.2307/2317088
Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983, 92(1):207–223. 10.1016/0022-247X(83)90280-9
Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471–473. 10.1007/BF01200091
Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986, 47(5):422–426. 10.1007/BF01189983
Alzer H, Qiu S-L: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201–215. 10.1007/s00013-003-0456-2
Acknowledgment
This work was supported by the Natural Science Foundation of Zhejiang Broadcast and TV University (Grant no. XKT-09G21).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Qian, WM., Zheng, NG. An Optimal Double Inequality for Means. J Inequal Appl 2010, 578310 (2010). https://doi.org/10.1155/2010/578310
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/578310