Schur Convexity and Inequalities for a Multivariate Symmetric Function

Abstract

In the article, we provide the Schur, Schur multiplicative, and Schur harmonic convexities properties for the symmetry function on and find several new analytical inequalities by use of the majorization theory, where , and are positive integers.

1. Introduction

Throughout the full article, we denote by the -dimensional Euclidean space, and . For and , we denote

Moreover, if , then we denote

Let be a nonempty interval. Then a real-valued function is said to be convex (concave) on if the inequality holds for all and .

It is well known that the convex (concave) function is one of the most important functions in geometric function theory, and it has wide applications in mathematics and physics as well as in the fields of engineering technology [1–5]. Recently, the generalizations, extensions, and variants for the convexity (concavity) have attracted the attention of many researchers, for example, the the - and -convexities [6], -convexity [7, 8], -convexity [9], -convexity [10], quasi-convexity [11], harmonic convexity [12, 13], exponential convexity [14], and generalized convexity [15]. In particular, many inequalities in pure and applied mathematics can be found in the literature [16–38] via the convexity (concavity) theory.

As one of the variants of the convexity, the Schur convexity was introduced by Schur in 1923, and it becomes the subject of current research. In [39], Xia and Chu discussed the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity for the -dimensional symmetry function on and established several interesting inequalities by use the majorization theory, where are positive integers.

From the definition of given in (4), we clearly see that the key item in for . But in many practical problems, what we need is that the main term . If we replace with , then we clearly see that for .

Motivated by the ideas given in [39] and the discussions above mentioned, we define the following symmetric function discuss its Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity properties on , and give their applications in the inequalities theory.

2. Definitions and Lemmas

To increase readability, we need recall some definitions in the beginning of this section. In order to establish our main results in the next section, we need introduce and establish several lemmas which we present in this section.

Definition 1 (see [40]). Let be a nonempty set. Then, a real-valued function defined on is said to be Schur convex if for each pair of -tuples such that , that is where is the th largest component of . is called Schur concave if is Schur convex.

Definition 2 (see [40]). Let be a nonempty set. Then a real-valued function defined on is said to be Schur multiplicatively convex if for each pair of -tuples such that . is called Schur multiplicatively concave if is Schur multiplicatively convex.

Definition 3 (see [40]). Let be a nonempty set. Then a real-valued function defined on is said to be Schur harmonic convex if for each pair of -tuples such that . is called Schur harmonic concave if inequality (9) is reversed.

Lemma 4 (see [40]). Let be a symmetric convex set with nonempty interior and be a continuous symmetry function on such that is differentiable on . Then is Schur convex on if and only if the inequality holds for and all . And is Schur concave on if and only if inequality (10) is reversed. Here, being a symmetric function in means that for any and any permutation matrix .

Remark 5. Since is symmetric, the Schur’s condition in Lemma 4, i.e., inequality (10) can be reduced to

Lemma 6 (see [40]). Let be a symmetric multiplicatively convex set with nonempty interior and be a continuous symmetry function on such that is differentiable in . Then is Schur multiplicatively convex on if and only if the inequality is valid for all . And is Schur multiplicatively concave on if and only if inequality (12) is reversed. Here, being a multiplicatively convex set means that for .

Lemma 7 (see [40]). Let be a symmetric harmonic convex set with nonempty interior and be a continuous symmetry function on such that is differentiable on . Then is Schur harmonic convex on if and only if the inequality takes place for all . And is Schur harmonic concave in if and only if inequality (13) is reversed. Here, being a harmonic convex set means that whenever .

Lemma 8 (see [40]). Let , and . Then,

Lemma 9 (see [40]). Let , and . Then,

Lemma 10 (see [40]). Let , and . Then,

Lemma 11. Let . Then, for .

Proof. Let . Then, we clearly see that for and for , which lead to the conclusion that is decreasing on and increasing on with respect to each . Therefore, for , and for , that is for .

Similarly, we can easily derive Lemma 12.

Lemma 12. Let and . Then, for .

3. Main Results

Theorem 13. The symmetric function defined by (5) is Schur convex on if and .

Proof. According to Lemma 4 and Remark 5, we only need to prove that for all , and .

We divide the proof into four cases.

Case 1. If , then it follows from (5) that and From (19), we clearly see that

Case 2. If and , then from (5), we obtain and Equation (22) leads to the conclusion that

Case 3. If and , then from (5), we get and for .
Equation (25) leads to It follows from that , , and . Let . Then, it is easy to check that for , and from Lemma 11, we know that for . Therefore, for follows from (26) immediately.

Case 4. If and , then (5) leads to and for .
Equation (28) leads to the conclusion that

Let . Then, we only need to prove that for . We divide the proof into three subcases.

Subcase 1. If , then it is easy to check that .

Subcase 2. If , then is increasing on , and from Lemma 12, we know that

Subcase 3. If , then is increasing on and we get From Subcases 1–3, we know that for all , and then from (29) we obtain for .
Cases 1–4 lead to the conclusion that inequality (17) holds for all and . Therefore, is Schur convex on which follows from Lemma 4 and Remark 5.

Remark 14. From the proof of Cases 1 and 2 in Theorem 13, we know that is Schur concave on and is Schur convex on .

Theorem 15. The symmetric function defined by (5) is Schur multiplicatively convex on if and .

Proof. According to Lemma 6, we only need to prove that for and . We divide the proof into four cases.

Case 1. If , then (19) leads to

Case 2. If and , then from (22), we get

Case 3. If and , then from (25), we have Making use of the arithmetric-geometric mean inequality, and the fact that for , we get for .
Note that , , and for . Therefore, follows from (35).

Case 4. If and , then it follows from (28) that Therefore, inequality (32) follows from Cases 1–4, and Theorem 15 follows from Lemma 6 and (32).

Theorem 16. The symmetric function given in (5) is Schur harmonic convex on if and .

Proof. According to Lemma 7, we only need to prove that for and . The proof is divided into four cases.

Case 1. If , then from (19), we get

Case 2. If and , then it follows from (22) that

Case 3. If and , then from (25), we obtain Applying (41) and the arithmetric-geometric mean inequality together with the facts that and for , we get .

Case 4. If and , then Equation (28) leads to From Cases 1–4, we know that inequality (38) holds for and .

4. Applications

In this section, we establish some new inequalities by the use of Theorems 13, 15, and 16 and the majorization theory.

Theorem 17. Let , , , , , , and . Then the following statements are true: (1)If , thenand (2)If and , then all the inequalities (44)–(47) are reversed; if and , then all the inequalities (44)–(47) also hold

Proof. Theorem 17 follows from Theorems 13, Remark 14, and Lemmas 8–10 together with the fact that

Theorem 18. Let , , and . Then, one has as follows: (1)If , then(2)If and , then the inequality (49) is reversed; if and , then the inequality (49) is also valid.

Proof. It is easy to see that Therefore, Theorem 18 follows from Theorem 13, Remark 14, and (50).

Remark 19. Let . Then Theorem 17 reduces to Theorem 18. On the other hand, if we take , then Theorem 18 gives the well-known Shapiro inequality [41]: for . If we take , then Theorem 18(2) leads to the well-known Ky Fan inequality [42]: for .

Theorem 20. If , , , and , then

Proof. We clearly see that Therefore, Theorem 20 follows from Theorem 15 and (54).
Let and . Then Theorem 20 leads to Corollary 21.

Corollary 21. If , and , then one has and

Theorem 22. If , , , and , then we have

Proof. It is easy to see that Therefore, Theorem 22 follows from Theorem 16 and (58) immediately.

Theorem 23. Suppose that is a complex matrix, and are the eigenvalues of . If is a positive definite Hermitian matrix, and is the unit matrix, then the inequalities and hold for all .

Proof. It is not difficult to verify that and Therefore, inequalities (59) and (60) follow from Theorems 15, (62), and (63), and inequality (61) follows from Theorem 16 and (64).

Theorem 24. Let be an -dimensional simplex of , be an arbitrary point in the interior of , and be the intersection point of straight line and hyperplane . Then, the inequalities and hold for all .

Proof. It is easy to see that and for , which imply that and Therefore, inequality (65) follows from (67), (68), and Theorem 13, and inequality (66) follows from (69), (70), and Theorem 13.

Remark 25. Mitrinović et al. [41] established a series of inequalities for and . Obviously, our inequalities in Theorem 24 are different from theirs.

5. Conclusions

We have presented the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity properties for the symmetric function defined by (5). As applications, we have found several new analytical inequalities by the use of the majorization theory. In particular, we have provided a common generalization for the Sharpiro and the Ky Fan inequalities by the use of our obtained results. Our results are the generalizations and improvements of the previously known results, and our ideas and approach may lead to a lot of follow-up research.

Data Availability

Not applicable..

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions. The work was supported by the Aid program for Science and Technology Innovative Research Team in Higher Educational Instituions of Hunan Province, the National Nature Science Foundation of China (Grant No. 11271118), and the Nature Science Foundation of Hunan Province (Grant No. 12JJ3002).