- Research Article
- Open Access
The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications
Journal of Inequalities and Applications volume 2009, Article number: 838529 (2009)
We prove that the Hamy symmetric function is Schur harmonic convex for . As its applications, some analytic inequalities including the well-known Weierstrass inequalities are obtained.
Throughout this paper we use to denote the -dimensional Euclidean space over the field of real numbers, and .
For , and , we denote by
Corresponding to this is the r th order Hamy mean
where . Hara et al.  established the following refinement of the classical arithmetic and geometric means inequality:
Here and denote the classical arithmetic and geometric means, respectively.
The paper  by Ku et al. contains some interesting inequalities including the fact that is log-concave, the more results can also be found in the book  by Bullen. In , the Schur convexity of Hamy's symmetric function and its generalization were discussed. In  , Jiang defined the dual form of the Hamy symmetric function as follows:
discussed the Schur concavity Schur convexity of , and established some analytic inequalities.
The main purpose of this paper is to investigate the Schur harmonic convexity of the Hamy symmetric function . Some analytic inequalities including Weierstrass inequalities are established.
2. Definitions and Lemmas
Schur convexity was introduced by Schur in 1923 , and it has many important applications in analytic inequalities [7–12], linear regression , graphs and matrices , combinatorial optimization , information-theoretic topics , Gamma functions , stochastic orderings , reliability , and other related fields.
For convenience of readers, we recall some definitions as follows.
A set is called a convex set if whenever . A set is called a harmonic convex set if whenever .
It is easy to see that is a harmonic convex set if and only if is a convex set.
Let be a convex set a function is said to be convex on if for all . Moreover, is called a concave function if is a convex function.
Let be a harmonic convex set a function is called a harmonic convex (or concave, resp.) function on if for all .
Definitions 2.2 and 2.3 have the following consequences.
If is a harmonic convex set and is a harmonic convex function, then
is a concave function. Conversely, if is a convex set and is a convex function, then
is a harmonic concave function.
Let be a set a function is called a Schur convex function on if
for each pair of -tuples and in , such that , that is,
where denotes the th largest component in . is called a Schur concave function on if is a Schur convex function on .
Let be a set a function is called a Schur harmonic convex (or concave, resp.) function on if
for each pair of and in , such that .
Definitions 2.4 and 2.5 have the following consequences.
Let be a set, and , then is a Schur harmonic convex (or concave, resp.) function on if and only if is a Schur concave (or convex, resp.) function on .
The notion of generalized convex function was first introduced by Aczél in . Later, many authors established inequalities by using harmonic convex function theory [21–28]. Recently, Anderson et al.  discussed an attractive class of inequalities, which arise from the notation of harmonic convex functions.
The following well-known result was proved by Marshall and Olkin .
Theorem 2 A.
Let be a symmetric convex set with nonempty interior , and let be a continuous symmetric function on . If is differentiable on , then is Schur convex (or concave, resp.) on if and only if
for all and . Here, is a symmetric set means that implies for any permutation matrix .
Since is symmetric, the Schur's condition in Theorem , that is, (2.6) can be reduced to
The following Lemma 2.7 can easily be derived from Fact, Theorem and Remark 2.6 together with elementary computation.
Let be a symmetric harmonic convex set with nonempty interior , and let be a continuous symmetry function on . If is differentiable on , then is Schur harmonic convex (or concave, resp.) on if and only if
for all .
Next we introduce two lemmas, which are used in Sections 3 and 4.
Lemma 2.8 ([5, page 234]).
For , if th r -th order symmetric function is defined as
Lemma 2.9 ([2, Lemma 2.2]).
Suppose that and . If , then
3. Main Result
In this section, we give and prove the main result of this paper.
The Hamy symmetric function is Schur harmonic convex in .
By Lemma 2.7, we only need to prove that
To prove (3.1), we consider the following possible cases for .
Case 1 ().
Then (1.2) leads to , and (3.1) is clearly true.
Case 2 ().
Then (1.2) leads to the following identity:
and therefore, (3.1) follows from (3.2).
Case 3 ().
Then (1.2) leads to
Simple computation yields
From (3.4) we get
Therefore, (3.1) follows from (3.5) and the fact that is increasing in .
Case 4 ().
Fix and let and We have the following identity:
Differentiating (3.6) with respect to and , respectively, and using Lemma 2.8, we get
From (3.7) we obtain
Therefore, (3.1) follows from (3.8) and the fact that is increasing in .
In this section, making use of our main result, we give some inequalities.
Suppose that with . If and , then
The proof follows from Theorem 3.1 and Lemma 2.9 together with (1.2).
If taking and in Theorem 4.1, respectively, then we have the following corollaries.
Suppose that with . If , then
Suppose that with . If , then
Taking in Corollaries 4.2 and 4.3, respectively, we get the following.
If and , then
Corollary 4.5 (Weierstrass inequalities [30, Page 260]).
If , and , then
If and , then
Let , and be the -tuple, then obviously
Therefore, Theorem 4.6 follows from Theorem 3.1, (4.7) ,and (1.2).
Let be an -dimensional simplex in -dimensional Euclidean space , and be the set of vertices. Let be an arbitrary point in the interior of A. If is the intersection point of the extension line of and the -dimensional hyperplane opposite to the point , and , then one has
It is easy to see that
(4.9) implies that
Therefore, Theorem 4.7 follows from Theorem 3.1, (4.10), and (1.2).
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This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y607128.