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
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.
Definitions 2.2 and 2.3 have the following consequences.
is a harmonic concave function.
Definitions 2.4 and 2.5 have the following consequences.
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.
The following Lemma 2.7 can easily be derived from Fact, Theorem and Remark 2.6 together with elementary computation.
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]).
3. Main Result
In this section, we give and prove the main result of this paper.
and therefore, (3.1) follows from (3.2).
Simple computation yields
From (3.7) we obtain
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).
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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