About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 674567, 5 pages
http://dx.doi.org/10.1155/2013/674567
Research Article

A Reverse Analytic Inequality for the Elementary Symmetric Function with Applications

1Department of Electronic Information, Teacher's College, Beijing Union University, Beijing 100011, China
2Basic Courses Department, Beijing Union University, Beijing 100101, China

Received 31 October 2013; Accepted 17 December 2013

Academic Editor: Fernando Simões

Copyright © 2013 Huan-Nan Shi and Jing Zhang. 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.

Abstract

We give a reverse inequality involving the elementary symmetric function by use of the Schur harmonic convexity theory. As applications, several new analytic inequalities for the -dimensional simplex are established.

1. Introduction

Let . The elementary symmetric functions are defined by , and for or .

In 1997, Leng and Tang [1] in order to achieve higher dimensional generalization of the famous Pedoe inequality [25] presented the following analytic inequality.

Let , and . Then

Chen et al. [6] gave an analytic proof which is skillful. By the method of successive adjustment, Fan [7] gave a simple elementary proof for inequality under conditions , , and . Using the same method, Shi [8] extended inequality (2) to the cases of for , where .

Theorem A. Let , and . Then for , one has where .

And Shi also pointed out that inequality (3) does not hold for .

In 2002, using analytical methods, Ma and Pu [9] gave parameter extension of the inequality (3).

Theorem B. Let , , , , and . Then for , one has

In 2005, Li et al. [10] gave an elementary proof for Theorem B when .

In this paper, by studying the Schur harmonic convexity of , we obtain a reverse of inequality (4) and accordingly establish the reverse of the corresponding simplex inequalities.

Our main result is the following theorem.

Theorem 1. Let , , , , . Then for , is Schur harmonically concave on . When , one has where is the harmonic mean of .

Taking in Theorem 1, we infer the following corollary.

Corollary 2. Let , , , and . Then for , one has where is the harmonic mean of .

Taking in Corollary 2, we obtain the following inequality.

Corollary 3. Let , , and . Then for , one has where is the harmonic mean of .

2. Definitions and Lemmas

Definition 4 (see [11, 12]). Let and .(i) is said to be majorized by (in symbols ) if for and , where and are rearrangements of and in a descending order.(ii) is called a convex set if for every and , where and with .(iii)Let . The function : is said to be a Schur convex function on if on implies . is said to be a Schur concave function on if and only if is Schur convex.

Lemma 5 (see [11, 12]). Let . Then where is the arithmetic mean of .

Definition 6 (see [11, 12]). Let . (i) is called a symmetric set, if implies for every permutation matrix .(ii)The function is called symmetric if for every permutation matrix , for all .

Lemma 7 (see [11, p. 84]). Let be symmetric and have a nonempty interior convex set , and let be continuous on and differentiable in . Then is the Schur-convex (Schur-concave) function if and only if is symmetric on and holds for any .

The Schur-convexity described the ordering of majorization; the order-preserving functions were first comprehensively studied by Issai Schur in 1923. It has important applications in analytic inequalities, combinatorial optimization, quantum physics, information theory, and so on. See [11].

In 2009, Chu and Lv [13] introduced the notion of harmonically Schur convex function and some interesting inequalities were obtained. The Schur harmonic convexity involving some special functions has been investigated; see, for example, [1420].

Definition 8 (see [13]). Let . (i)A set is said to be harmonically convex if for every and , where and .(ii)A function is said to be Schur harmonically convex on if implies . A function is said to be a Schur harmonically concave function on if and only if is a Schur harmonically convex function.

Lemma 9 (see [13]). Let be a symmetric and harmonically convex set with inner points and let be a continuously symmetric function which is differentiable on . Then, is Schur harmonically convex (Schur harmonically concave) on if and only if

Lemma 10. Let , , , , , and . Then for , one has

Proof. Let . Then ; and , from Theorem B, we have and then Lemma 10 is proved.

3. Proof of Main Result

Proof of Theorem 1. When , , obviously, inequality (5) holds. When , , noting that we have And then
Now we distinguish two cases to prove .
Case 1. If , , then from the definition of elementary symmetric functions, it follows that .
If , , then from the condition , it follows that , and by , it follows that , ; therefore,
Case 2. If ; then , from , it follows that and . So by Lemma 10, it follows that .
Thus, for all , . By Lemma 9, we can derive that is Schur harmonically concave on .
From Lemma 5, it is seen that According to Definition 8(ii), inequality (5) follows.
The proof of Theorem 1 is completed.

4. Applications in Geometry

Let be an -dimensional simplex in the -dimensional Euclidean space with vertices whose volume is . For , let be the radius of th escribed sphere of , and the area of the th face of , . and are circumradius and inradius of , respectively. Let be an arbitrary interior point of the simplex , the distance from the point to the th face of , and the altitude of from vertex for .

In [9], by Theorem B, Ma and Pu obtained the following two theorems.

Theorem C. In the simplex (), for , one has with equality if and only if is an orthocentric simplex.

Theorem D. Let be an arbitrary point in the interior of -dimensional simplex . Let be the intersection of the line with the hyperplane ; further let , , , . Then for , one has with equality if and only if is an orthocentric simplex, and is its orthocentre.

Now, by Theorem 1, we give the reverse of inequalities (21) and (22).

Theorem 11. In the simplex , for , one has where .

Proof. In the -dimensional simplex, for , it is well known that
Let , . Then, . From equalities (24), we have
Thus from the inequality (6), it is deduced that the inequality (23) holds.
The proof of Theorem 11 is completed.

Theorem 12. Let be an arbitrary point in the interior of -dimensional simplex . Let be the intersection of the line with the hyperplane ; further let , , , . Then, for , where .

Proof. It is easy to see that
In the inequality (5), let , . Then, , , . Thus, inequality (26) holds.
The proof of Theorem 12 is completed.

When , inequality (26) is reduced to the following.

Corollary 13. Under the conditions of Theorem 12, one has

Theorem 14. In the simplex , let Then, where ; where .

Proof. Taking and applying Corollary 3 for these positive real numbers , inequality (30) follows.
Taking and applying Corollary 3, the desired inequality (31) follows.
The proof of Theorem 14 is completed.

Remark 15. Inequalities (30) and (31) give the reverse of the inequalities (2.10) and (2.11) in [10], respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (IDHT201304089) and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407). The authors are grateful to Professor Fernando Simões and the four anonymous reviewers for helpful pieces of advice that improved its presentation.

References

  1. G. S. Leng and L. H. Tang, “Some generalizations to several dimensional of the Pedoe inequality with applications,” Acta Mathematica Sinica, vol. 40, no. 1, pp. 14–21, 1997 (Chinese).
  2. L. Yang and J. Z. Zhang, “Generalization to higher dimensions of the Neuberg-Pedoe inequqlity with applitions,” Acta Mathematica Sinica, vol. 24, no. 3, pp. 401–408, 1981 (Chinese).
  3. L. Yang and J. Z. Zhang, “A generalisation to several dimensions of the Neuberg-Pedoe inequality, with applications,” Bulletin of the Australian Mathematical Society, vol. 27, no. 2, pp. 203–214, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  4. D. S. Mitrinović and J. E. Pečarić, “About the Neuberg-Pedoe and the Oppenheim inequalities,” Journal of Mathematical Analysis and Applications, vol. 129, no. 1, pp. 196–210, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  5. D. S. Mitrinović, J. E. Pečarić, and V. Volenec, Recent Advances in Geometric Inequalities, Kluwer Academic, Dordrecht, The Netherlands, 1989. View at MathSciNet
  6. J. Chen, Y. Huang, and S. H. Xia, “A note on generalizations of Neuberg-Pedoe inequality to higher dimensional case,” Journal of Sichuan University, vol. 36, no. 2, pp. 197–200, 1999 (Chinese).
  7. Y. W. Fan, “An generalization of Neuberg-Pedoe inequality for simplies,” Journal of Yanan University, vol. 18, no. 3, pp. 19–22, 1999 (Chinese).
  8. H. N. Shi, “The generalization of a analytic inequality,” Journal of Chengdu University, vol. 20, no. 1, pp. 5–7, 2001 (Chinese).
  9. T. Y. Ma and Z. N. Pu, “Some generalizations of an analytical inequality and their applications,” Journal of Sichuan University, vol. 39, no. 1, pp. 1–6, 2002 (Chinese). View at MathSciNet
  10. X. Y. Li, G. S. Leng, and L. H. Tang, “Inequalities for a simplex and any point,” Mathematical Inequalities & Applications, vol. 8, no. 3, pp. 547–557, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  11. A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Application, Springer, New York, NY, USA, 2nd edition, 2011.
  12. B. Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University, Beijing, China, 1990, (Chinese).
  13. Y. M. Chu and Y. P. Lv, “The Schur harmonic convexity of the Hamy symmetric function and its applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 838529, 10 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  14. W. F. Xia and Y. M. Chu, “The Schur harmonic convexity of Lehmer means,” International Mathematical Forum, vol. 4, no. 41–44, pp. 2009–2015, 2009. View at MathSciNet
  15. Y. M. Chu and T. C. Sun, “The Schur harmonic convexity for a class of symmetric functions,” Acta Mathematica Scientia B, vol. 30, no. 5, pp. 1501–1506, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Y.-M. Chu and W.-F. Xia, “Necessary and sufficient conditions for the Schur harmonic convexity of the generalized Muirhead mean,” Proceedings of A. Razmadze Mathematical Institute, vol. 152, pp. 19–27, 2010. View at MathSciNet
  17. Z.-H. Yang, “Schur harmonic convexity of Gini means,” International Mathematical Forum, vol. 6, no. 13–16, pp. 747–762, 2011. View at MathSciNet
  18. W.-F. Xia, Y.-M. Chu, and G.-D. Wang, “Necessary and sufficient conditions for the Schur harmonic convexity or concavity of the extended mean values,” Revista de la Unión Matemática Argentina, vol. 51, no. 2, pp. 121–132, 2010. View at MathSciNet
  19. H. N. Shi, Theory of Majorization and Analytic Inequalities, Harbin Institute of Technology, Harbin, China, 2012, (Chinese).
  20. H. N. Shi and J. Zhang, “Some new judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions,” Journal of Inequalities and Applications, vol. 2013, article 527, 2013.