New proofs of Schur-concavity for a class of symmetric functions

Shi Huan-Nan, Zhang Jian, Gu, Gu Chun

Description

By properties of the Schur-convex function, Schur-concavity for a class of symmetric functions is simply proved uniform. 2000 Mathematics Subject Classification : Primary 26D15; 05E05; 26B25. By properties of the Schur-convex function, Schur-concavity for a class of symmetric functions is simply proved uniform. 2000 Mathematics Subject Classification : Primary 26D15; 05E05; 26B25.

Sujets

• Majorization
• Inequality
• Symmetric function
• Concave function

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 6
• Langue: English

Extrait

Shiet al.Journal of Inequalities and Applications2012,2012:12 http://www.journalofinequalitiesandapplications.com/content/2012/1/12

R E S E A R C HOpen Access New proofs of Schurconcavity for a class of symmetric functions * HuanNan Shi , Jian Zhang and Chun Gu

* Correspondence: [e-mail] Department of Electronic Information, Teacher’s College, Beijing Union University, Beijing 100011, P.R. China

Abstract By properties of the Schurconvex function, Schurconcavity for a class of symmetric functions is simply proved uniform. 2000 Mathematics Subject Classification: Primary 26D15; 05E05; 26B25. Keywords:majorization, Schurconcavity, inequality, symmetric functions, concave functions

1. Introduction Throughout the article,ℝdenotes the set of real numbers,x= (x 1 , x 2 , ...,x n ) denotes ntuple (ndimensional real vectors), the set of vectors can be written as n R ={x= (x 1 , ...,x n ) :x i ∈ R ,i...,= 1,n}, n R ={x= (x 1 , ...,x n ) :x i >0,i...,= 1,n}. + 1 1 In particular, the notteℝand R spectively. ationsℝandℝ + deno + re For convenience, we introduce some definitions as follows. n Definition 1. [1,2] Letx =(x 1 , ..., x n ) andy =(y 1 , ..., y n )Îℝ.

(i)x≥ymeansx i ≥y i for alli= 1, 2,..., n. n (ii) LetΩ⊂ℝ,:Ω®ℝis said to be increasing ifx≥yimplies(x)≥(y).is said to be decreasing if and only ifis increasing.

n Definition 2. [1,2] Letx =(x 1 , ..., x n ) andy=(y 1 , ..., y n )Îℝ.

  k k (i)xis said to be mafk= jorized byy(in symbolsx≺y) if x [i] ≤y [i] or i=1i=1   n n 1, 2,..., n 1 and x i =y i , wherex [1] ≥∙ ∙ ∙≥x [n] andy [1] ≥∙ ∙ ∙≥y [n] are i=1i=1 rearrangements ofxandyin a descending order. n (ii) LetΩ⊂ℝ,:Ω®ℝis said to be a Schurconvex function onΩifx≺yon Ωimplies(x)≤(y).is said to be a Schurconcave function onΩif and only ifis Schurconvex function onΩ.

n Definition 3. [1,2] Letx=(x 1 , ..., x n ) andy= (y 1 , ..., y n )Îℝ.

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