A note on Schur-concave functions

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Description

In this paper we consider a class of Schur-concave functions with some measure properties . The isoperimetric inequality and Brunn-Minkowsky’s inequality for such kind of functions are presented. Applications in geometric programming and optimization theory are also derived. MSC: 26B25, 26B15, 52A40. In this paper we consider a class of Schur-concave functions with some measure properties . The isoperimetric inequality and Brunn-Minkowsky’s inequality for such kind of functions are presented. Applications in geometric programming and optimization theory are also derived. MSC: 26B25, 26B15, 52A40.

Sujets

• Isoperimetric inequality
• Optimization

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 12
• Langue: English

Extrait

Roven¸taJournal of Inequalities and Applications2012,2012:159 http://www.journalofinequalitiesandapplications.com/content/2012/1/159

R E S E A R C H

A note on Schur-concave functions * IonelRoven¸ta

* Correspondence: [e-mail] Department of Mathematics, University of Craiova, Craiova 200585, Romania

Open Access

Abstract In this paper we consider a class of Schur-concave functions with somemeasure properties. The isoperimetric inequality and Brunn-Minkowsky’s inequality for such kind of functions are presented. Applications in geometric programming and optimization theory are also derived. MSC:26B25; 26B15; 52A40 Keywords:Schur-concave functions; isoperimetric inequality; optimization

1 Introduction About  years ago, the properties concerning such notions as length, area, volume as well as the probability of events were abstracted under the banner of the wordmeasure. We review the notion of measure using this word in an unusual way. More exactly, we study some measure properties of a special class of Schur-concave functions which will be revealed via some discrete versions of isoperimetric inequality and Brunn-Minkowsky’s inequality. The notion of Schur-convex function was introduced by I. Schur in  and has had in- teresting applications in analytic inequalities, elementary quantum mechanics and quan- tum information theory. See []. Let us considerx= (x  , . . . ,x n ),y= (y  , . . . ,y n ) to be two n vectors fromR.

Definition We say thatxis majorized byy, denote it byx≺y, if the rearrangement of the components ofxandysuch thatx [] ≥x [] ≥ ∙ ∙ ∙ ≥x [n] ,y [] ≥y [] ≥ ∙ ∙ ∙ ≥y [n] satisfies    k kn n x y(≤k≤n– ) andx=y. i = [i] ≤ i = [i] i = [i] i = [i]

n Definition The functionF:A→R, whereA⊂R, is called Schur-convex ifx≺yim- pliesF(x)≤F(y). Any such functionFis called Schur-concave if –Fis Schur-convex.

An important source of Schur-convex functions can be found in Merkle []. Guan [, ] proved that all symmetric elementary functions and the symmetric means of order kare Schur-concave functions. Other families of Schur-convex functions are studied in [–, –, ]. In [] a class of analytic inequalities for Schur-convex functions that are made of so- lutions of a second order nonlinear differential equation was established. These analytic inequalities are used to infer some geometric inequalities such as isoperimetric inequal- ity. Li and Trudinger [] consider a special class of inequalities for elementary symmetric functions that are relevant to the study of partial differential equations associated with curvature problems.

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