- Research Article
- Open Access
The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications
© Y. Chu and Y. Lv. 2009
- Received: 2 April 2009
- Accepted: 20 May 2009
- Published: 28 June 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.
- Convex Function
- Symmetric Function
- Concave Function
- Nonempty Interior
- Dimensional Euclidean Space
Throughout this paper we use to denote the -dimensional Euclidean space over the field of real numbers, and .
Here and denote the classical arithmetic and geometric means, respectively.
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.
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.
is a harmonic concave function.
where denotes the th largest component in . is called a Schur concave function on if is a Schur convex function on .
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.
for all and . Here, is a symmetric set means that implies for any permutation matrix .
The following Lemma 2.7 can easily be derived from Fact, Theorem and Remark 2.6 together with elementary computation.
for all .
Next we introduce two lemmas, which are used in Sections 3 and 4.
Lemma 2.8 ([5, page 234]).
Lemma 2.9 ([2, Lemma 2.2]).
In this section, we give and prove the main result of this paper.
The Hamy symmetric function is Schur harmonic convex in .
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 ( ).
and therefore, (3.1) follows from (3.2).
Case 3 ( ).
Simple computation yields
Therefore, (3.1) follows from (3.5) and the fact that is increasing in .
Case 4 ( ).
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.
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.
Taking in Corollaries 4.2 and 4.3, respectively, we get the following.
Corollary 4.5 (Weierstrass inequalities [30, Page 260]).
Therefore, Theorem 4.6 follows from Theorem 3.1, (4.7) ,and (1.2).
Therefore, Theorem 4.7 follows from Theorem 3.1, (4.10), and (1.2).
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.
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