#### 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 , then**and
*(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]: