# A Note on Mixed-Mean Inequalities

- Peng Gao
^{1}Email author

**2010**:509323

https://doi.org/10.1155/2010/509323

© Peng Gao. 2010

**Received: **25 February 2010

**Accepted: **29 June 2010

**Published: **14 July 2010

## Abstract

We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed-mean inequality involving the symmetric means.

## Keywords

## 1. Introduction

Let be the generalized weighted power means: , where , , and , with . Here denotes the limit of as . Unless specified, we always assume that . When there is no risk of confusion, we will write for and we also define .

where are two matrices with nonnegative entries and the above inequality being meant to hold for any vector with nonnegative entries. Here and, when , we want the inequality above to be reversed.

Now we focus our attention to the case of (1.2) for being weighted mean matrices given in (1.3). In this case, for fixed , we define , . Then we have the following mixed-mean inequalities of Nanjundiah [3] (see also [4]).

Theorem 1.1.

with equality holding if and only if .

A very elegant proof of Theorem 1.1 for the case is given by Kedlaya in [5]. In fact, the following Popoviciu-type inequalities were established in [5] (see also [4, Theorem ]).

Theorem 1.2.

with equality holding if and only if .

It is easy to see that the case of Theorem 1.1 follows from Theorem 1.2. As was pointed out by Kedlaya that the method used in [5] can be applied to establish both Popoviciu-type and Rado-type inequalities for mixed means for a general pair , the details were worked out in [6] and the following Rado-type inequalities were established in [6].

Theorem 1.3.

with equality holding if and only if and the above inequality reverses when .

A different proof of Theorem 1.1 for the case was given in [7] and Bennett used essentially the same approach in [8, 9] to study (1.2) for the cases being lower triangular matrices, namely, if . Among other things, he showed [8] that inequalities (1.2) hold when are Hausdorff matrices.

In [10], Holland further improved the condition in Theorem 1.3 for the case by proving the following.

Theorem 1.4.

with equality holding if and only if .

It is our goal in this paper to first give a simpler proof of the above result by modifying Holland's own approach. This is done in the next section and, in Section 3, we will prove a result of mixed-mean inequality involving the symmetric means.

## 2. A Proof of Theorem 1.4

## 3. A Discussion on Symmetric Means

It is well known that, for fixed of dimension , is a nonincreasing function of for with (with weights , ). In view of the mixed-mean inequalities for the generalized weighted power means (Theorem 1.1), it is natural to ask whether similar results hold for the symmetric means. Of course one may have to adjust the notion of such mixed means in order for this to make sense for all . For example, when , , the notion of is not even defined. From now on, we will only focus on the extreme cases of the symmetric means; namely, or . In these cases it is then natural to define , and, on recasting , we see that it is also natural for us to define (note that this is not consistent with our definition of above).

We now prove a mixed-mean inequality involving and . We first note the following result of Marcus and Lopes [11] (see also [12, pages 33–35]).

Theorem 3.1.

with equality holding if and only if or there exists a constant such that .

We also need the following lemma of C. D. Tarnavas and D. D. Tarnavas [6].

Lemma 3.2.

The equality holds if and only if or when is strictly convex. When is concave, then the above inequality is reversed.

We now apply Lemma 3.2 to obtain the following.

Lemma 3.3.

with equality holding in both cases if and only if or .

Proof.

which is just what we want.

We now prove the following mixed-mean inequality involving the symmetric means.

Theorem 3.4.

with equality holding if and only if .

Proof.

where the last inequality follows from Theorem 3.1 for the case . It is easy to see that the above inequality is equivalent to (3.9) and this completes the proof.

where we define . We now end this paper by proving the following result.

Theorem 3.5.

Proof.

It is easy to see that the last expression above is no less than when and this proves inequality (3.18) for the case . This completes the proof of the theorem.

## Authors’ Affiliations

## References

- Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. Cambridge University Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar - Čižmešija A, Pečarić J: Mixed means and Hardy's inequality.
*Mathematical Inequalities & Applications*1998, 1(4):491–506.MathSciNetMATHGoogle Scholar - Nanjundiah TS: Sharpening of some classical inequalities.
*Math Student*1952, 20: 24–25.Google Scholar - Bullen PS: Inequalities due to T. S. Nanjundiah. In
*Recent Progress in Inequalities*.*Volume 430*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:203–211.View ArticleGoogle Scholar - Kedlaya KS: A weighted mixed-mean inequality.
*The American Mathematical Monthly*1999, 106(4):355–358. 10.2307/2589560MathSciNetView ArticleMATHGoogle Scholar - Tarnavas CD, Tarnavas DD: An inequality for mixed power means.
*Mathematical Inequalities & Applications*1999, 2(2):175–181.MathSciNetView ArticleMATHGoogle Scholar - Kedlaya K: Proof of a mixed arithmetic-mean, geometric-mean inequality.
*The American Mathematical Monthly*1994, 101(4):355–357. 10.2307/2975630MathSciNetView ArticleMATHGoogle Scholar - Bennett G: An inequality for Hausdorff means.
*Houston Journal of Mathematics*1999, 25(4):709–744.MathSciNetMATHGoogle Scholar - Bennett G: Summability matrices and random walk.
*Houston Journal of Mathematics*2002, 28(4):865–898.MathSciNetMATHGoogle Scholar - Holland F: An inequality between compositions of weighted arithmetic and geometric means.
*Journal of Inequalities in Pure and Applied Mathematics*2006, 7, article 159:-8.Google Scholar - Marcus M, Lopes L: Inequalities for symmetric functions and Hermitian matrices.
*Canadian Journal of Mathematics*1957, 9: 305–312. 10.4153/CJM-1957-037-9MathSciNetView ArticleMATHGoogle Scholar - Beckenbach EF, Bellman R:
*Inequalities*. Springer, Berlin, Germany; 1961:xii+198.View ArticleMATHGoogle Scholar - Knopp K: Über Reihen mit positiven Gliedern.
*Journal of the London Mathematical Society*1928, 3: 205–211. 10.1112/jlms/s1-3.3.205MathSciNetView ArticleMATHGoogle Scholar - Knopp K: Über Reihen mit positiven Gliedern (Zweite Mitteilung).
*Journal of the London Mathematical Society*1930, 5: 13–21. 10.1112/jlms/s1-5.1.13MathSciNetView ArticleMATHGoogle Scholar - Duncan J, McGregor CM: Carleman's inequality.
*The American Mathematical Monthly*2003, 110(5):424–431. 10.2307/3647829MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.