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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 258108, 12 pages
http://dx.doi.org/10.1155/2014/258108
Research Article

Schur -Power Convexity of a Class of Multiplicatively Convex Functions and Applications

1School of Mathematics and Statistics, Hefei Normal University, Hefei 230601, China
2Department of Mathematics and Physics, Anhui Xinhua University, Hefei 230088, China

Received 14 January 2014; Revised 2 April 2014; Accepted 23 April 2014; Published 20 May 2014

Academic Editor: Marco Sabatini

Copyright © 2014 Wen Wang and Shiguo Yang. 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 investigate the conditions under which the symmetric functions are Schur -power convex for and . As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving the th power mean and the arithmetic, the geometric, or the harmonic means are presented.

1. Introduction

Let and . In particular, . For , we denote by

The Hamy symmetric function [1, 2] is defined as

The properties and applications of Hamy symmetric function can be found in the book of Bullen et al. [1].

Throughout the paper, let and .

In 2007, Guan [3] defined a more general symmetric function: let , and where are nonnegative integers, . Guan proved that the above symmetric function is Schur geometrically convex on .

In 2010, Rovenţa [4] defined the following symmetric function. Let be a log-convex function

Rovenţa proved that (4) is a Schur convex function on .

In 2010, Meng et al.  [5] proved the dual form of the Hamy symmetric function was Schur harmonically convex in .

In 2013, Shi and Zhang [6] investigated the following dual form of : They proved that is Schur convex, Schur geometrically and harmonically convex on .

Recently, Yang [79] generalized the notion of Schur convexity to Schur -power convexity, which contains the Schur convexity, Schur geometrical convexity, and Schur harmonic convexity. Moreover, he discussed Schur -power convexity of Stolarsky means [7], Gini means [8], and Daróczy means [9]. Wang and Yang showed that generalized Hamy symmetric function [10] and a class of symmetric functions [11] are Schur -power convex.

Now we define the more general dual form of symmetric function.

Definition 1. Let be a symmetric convex set with nonempty interior and is continuous on and differentiable in the interior of . For , define the symmetric functions by

In this paper, we investigate the Schur -power convexity of the above more general dual form of symmetric functions. In particular, we obtain that the above more general dual form of symmetric functions is Schur geometrically convex and Schur harmonically convex, which generalizes some known results. As a consequence, we are able to prove a number of new inequalities concerning the th power mean, the arithmetic mean, and the geometric and the harmonic mean.

2. Definitions and Lemmas

We first recall several definitions as follows.

Definition 2 (see [12, 13]). Let and .(1) means for all .(2)Let , is said to be increasing if implies . is said to be decreasing if and only if is increasing.

Definition 3 (see [12, 13]). Suppose that and are two -tuples real numbers.(1) majorizes (in symbols ), if and , where are rearrangements of and in a descending order.(2)A real-valued function is said to be Schur convex on if is a Schur concave function on if and only if is a Schur convex function.

Definition 4 (see [14]). Suppose and are two -tuples real numbers. Let . A function is called Schur geometrically convex if is Schur geometrically concave if is Schur geometrically convex.

The following Theorem is basic and plays an important role in the theory of the Schur geometrically convex function.

Lemma 5 (see [14]). Let be symmetric and continuous on and differentiable in . Then is Schur geometrically convex (Schur geometrically concave) if and only if

Definition 6 (see [15, 16]). Let .(1)A set is called harmonically convex if for every and , where and .(2)A function is called Schur harmonically convex on if implies . is Schur harmonically concave if is Schur harmonically convex.

Lemma 7 (see [15, 16]). Let be a symmetric and harmonically convex set with inner points and let be a continuously symmetric function which is differentiable in . Then is Schur harmonically convex (Schur harmonically concave) on if and only if Schur convex, Schur geometrically convex, and Schur harmonically convex were introduced by Marshall et al. [13], Zhang [14], and Chu and Sun [15], respectively, and played a key role in analytic inequalities [136]. Moreover, the theory of convex functions and Schur convex functions is one of the most important research fields in modern analysis and geometry.

Recently, Yang presents the Schur f-convexity in [7] as follows.

Definition 8 (see [79]). Let be a set with nonempty interior and a strictly monotone function defined on . Let Then function is said to be Schur f-convex on if on implies .
is said to be Schur f-concave if is Schur f-convex.

Take , , in Definition 8, it yields the Schur convexity, Schur geometrical convexity, and Schur harmonic convexity. It is clear that the Schur f-convexity is a generalization of the Schur convexity mentioned above. In general, we have the following.

Definition 9 (see [79]). Let be defined by if and if . Then function is said to be Schur -power convex on if on implies .
is said to be Schur -power concave if is Schur -power convex.

Lemma 10 (see [79]). Let be continuous on and differentiable in . Then is schur -power convex (Schur -power concave) on if and only if is symmetric on and hold for any with , where is a symmetric set with nonempty interior .

The following lemma is clearly due to the monotonicity property of the function on .

Lemma 11 (see [10]). For with , let be defined by
Then , (i.e., .)

Remark 12 (see [10]). By Lemma 11, we see that Then the two discrimination inequalities in Lemma 10 are equivalent to

Definition 13 (see [3, 17]). Function is said to be multipicatively convex if

The following results have been proven, respectively.

Lemma 14 (see [17]). A continuous function is multiplicatively convex if and only if or

Lemma 15 (see [17]). Assume that is a differential function. Then the following assertions are equivalent. (i)f is multiplicatively convex.(ii)The function is nondecreasing.

Moreover, if is twice differentiable, then is multiplicatively convex if and only if

3. Main Results and Proof

Our main results are stated as follows.

Theorem 16. If is increasing and multiplicatively convex, then for and , defined in (7) are Schur -power convex on , where .

Take in Theorem 16, we get the following Corollaries.

Corollary 17. If is increasing and multiplicatively convex, then for any and , defined in (7) are Schur geometrically convex on .

Corollary 18. If is increasing and multiplicatively convex, then for any and , defined in (7) are Schur harmonically convex function on .

Theorem 19. Let , and . If is increasing and multiplicatively convex, then for any and , one has

To prove the main results, we first establish some lemmas.

Lemma 20. Let the function be continuous on and differentiable in the interior of . For , if is increasing and multiplicatively convex, then is increasing.

Proof. Since is multiplicatively convex, and by using Lemma 15, we can easily see that is increasing. Further, by applying and the monotonicity of , it follows that is also increasing for and .

Lemma 21. If , and , the function is increasing.

Proof. We can easily derive that
So the function is increasing.

Lemma 22. Let be continuous on and differentiable in the interior of . For and , if is increasing and multiplicatively convex, then

Proof. Firstly, we prove that (23) holds. Since and , then So, we deduce that .
Secondly, we prove that (24) holds. Set ,, obviously, . One can easily find that Because is multiplicatively convex, and by Lemma 15, we get
On the other hand, for and , we easily know that the functions and are increasing about . By applying Lemma 21, we have
Because the function is increasing and , and applying (27)-(28), we obtain .

Lemma 23. Let be continuous on and differentiable in the interior of . If is increasing and multiplicatively convex, then for and , one has

Proof. Set , ; it is easy that . Then where .
By (27)-(28) and the monotonicity property of the nonnegative function , we get that .

Proof of Theorem 16. By Lemma 10 and Remark 12, we only need to prove that
To prove the above inequality, we consider the following three cases for .
Case 1. For . It is clear that . From (31), it follows that
By Lemma 20, and , it follows that .
Case 2. For , we have . We can easily derive that By differentiating the above equation with respect to , we obtain
Similarly, we have
So, from (34) and (35), and by applying Lemma 22, we have
So we get that .
Case 3. For , similarly to the discussion of Case 2, we have
By differentiating to the above with respect to , we have
Similarly, we can have
From (38) and (39), we have
So we get that . So the proof of Theorem 16 is complete.

Proof of Theorem 19. Let , and . From [13, 14], we have
From (41), and contacting Corollaries 17 and 18, we get (21).

4. Applications

Let be positive real numbers and set . The th power mean of of order of is defined by

In particular, for , and we, respectively, get the arithmetic, the geometric, and the harmonic means of , and set , ,

In this section, some applications of the results in Section 3 are given. Some analytic inequalities are established. In particular, several inequalities involving the th power mean and the arithmetic, the geometric, or the harmonic means are presented.

4.1. Analytic Inequalities

To establish some analytic inequalities, we first give a lemma.

Lemma 24 (see [20]). Let . Then the following relations is known:

By applying Lemma 24 and Theorem 16, Corollaries 17 and 18, we can get the following several theorems and corollaries.

Theorem 25. For and , the symmetric functions are Schur -power convex in when . In particular, are Schur geometrically convex and Schur harmonically convex in .

Proof. For , let . Then and Let ; then , and is increasing. From the monotonicity of , we get that Therefore, is increasing and multiplicatively convex in . By applying Theorem 16, we obtain the result.

By using Theorem 25 and Lemma 24, we get the following inequalities.

Corollary 26. For , then

Theorem 27. For and , the symmetric functions are Schur -power convex in when . In particular, are Schur geometrically convex and Schur harmonically convex in .

Proof. For , let . Then and Let ; then , and is increasing. From the monotonicity of , we get that Therefore, is increasing and multiplicatively convex in . By applying Theorem 16, we obtain the result.

By using Theorem 27 and Lemma 24, we get the following inequalities.

Corollary 28. For , then

Theorem 29. For and , the symmetric functions are Schur -power convex in when . In particular, are Schur geometrically convex in and Schur harmonically convex in .

Proof. Let . We can easily see that is increasing and multiplicatively convex in . By applying Theorem 16, we obtain the result.

By using Theorem 29 and Lemma 24, we obtain the following results.

Corollary 30. Let , for and ; one has

In particular, take in (53); we get the known A-G-H inequality. And take in (53); we get the following inequalities:

Theorem 31. For and , the symmetric functions are Schur -power convex on when . In particular, are Schur geometrically convex on and Schur harmonically convex on .

Proof. Let . Then , and So is increasing and multiplicatively convex in . By applying Theorem 16, we obtain the result.

By using Theorem 31 and Lemma 24, we obtain the following results.

Corollary 32. Let , for and , one has

In particular, take or in (59), we get the following inequalities:

Theorem 33. For and , the symmetric functions are Schur -power convex on when . In particular, are Schur geometrically convex and Schur harmonically convex on .

Proof. Let . Then , and So is increasing and multiplicatively convex in . By applying Theorem 16, we obtain the result.

By using Theorem 33 and Lemma 24, we get the following inequalities.

Corollary 34. Let , for and ; one has

In particular, take or in (63); we get the following inequalities:

Theorem 35. For and , the symmetric functions are Schur -power convex on when . In particular, are Schur geometrically convex and Schur harmonically convex on .

Proof. Let . Then , and So is increasing and multiplicatively convex on . By applying Theorem 16, we obtain the result.

By using Theorem 35 and Lemma 24, we get the following inequalities.

Corollary 36. Let , for and ; one has

In particular, take or in (67); we get the following inequalities:

4.2. Geometric Inequalities

In this section, some geometric inequalities of -dimensional simplex are established by use of the results of Theorem 16.

Lots of geometric inequalities for an -dimensional simplex are established (see [3742]). In this section, applying the above Lemma and the main results in Section 2, we establish some interesting geometric inequalities on -dimensional simplex in -dimensional Euclidean space .

In what follows, Let be an -dimensional simplex in -dimensional Euclidean space    with the volume. We denote by , , , and the altitudes, the radii of excircles, the areas of lateral surfaces, and the inradius of , respectively. For a given point in , let stand for the intersection point of straight line and hyperplane .

We first give some lemmas.

Lemma 37. Let be an -dimensional simplex in -dimensional Euclidean space   . The following relation is where .

Proof. By the formula [37, 38]