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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 674567, 5 pages
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.
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.
Let . The elementary symmetric functions are defined by , and for or .
Let , and . Then
Chen et al.  gave an analytic proof which is skillful. By the method of successive adjustment, Fan  gave a simple elementary proof for inequality under conditions , , and . Using the same method, Shi  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 .
Theorem B. Let , , , , and . Then for , one has
In 2005, Li et al.  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 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 .
In 2009, Chu and Lv  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, [14–20].
Definition 8 (see ). 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 ). 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
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 , 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.
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 .
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.
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