Schur-Convexity for a Class of Symmetric Functions and Its Applications
© W.-F. Xia and Y.-M. Chu. 2009
Received: 16 May 2009
Accepted: 14 September 2009
Published: 12 October 2009
For the sake of convenience, we use the following notation system.
The notion of Schur convexity was first introduced by Schur in 1923 . It has many important applications in analytic inequalities [2–7], combinatorial optimization , isoperimetric problem for polytopes , linear regression , graphs and matrices , gamma and digamma functions , reliability and availability , and other related fields. The following definition for Schur convex or concave can be found in [1, 3, 7] and the references therein.
In paper , Anderson et al. discussed an attractive class of inequalities, which arise from the notion of harmonic convex functions. Here, we introduce the notion of Schur harmonic convexity.
The main purpose of this paper is to discuss the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of the following symmetric function:
In order to establish our main results we need several lemmas, which we present in this section.
The following lemma is so-called Schur's condition which is very useful for determining whether or not a given function is Schur convex or Schur concave.
This fact, Lemma 2.1 and Remark 2.2 together with elementary calculation imply that Lemma 2.4 is true.
Lemma 2.6 (see ).
Lemma 2.7 (see ).
3. Main Results
The proof is divided into four cases.
Therefore, (3.1) follows from Cases 1–4 and the proof of Theorem 3.1 is completed.
Therefore, Theorem 3.2 follows from (3.10) and Cases 1–4.
The proof is divided into four cases.
Therefore, (3.15) follows from Cases 1–4 and the proof of Theorem 3.4 is completed.
In this section, we establish some inequalities by use of Theorems 3.1, 3.2 and 3.4 and the theory of majorization.
Therefore, Theorem 4.7 follows from (4.10), Theorem 3.2, and (1.7).
Therefore, Theorem 4.11 follows from (4.15), Theorems 3.1, 3.4, and (1.7).
Mitrinovi et al. [21, pages 473–479] established a series of inequalities for and , . Obvious, our inequalities in Theorem 4.11 are different from theirs.
This work was supported by the National Science Foundation of China (no. 60850005) and the Natural Science Foundation of Zhejiang Province (no. D7080080, Y607128, Y7080185).
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