Research Article Schur-Convexity for a Class of Symmetric Functions and Its Applications

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1 Hindawi Publishing Copoation Jounal of Inequalities and Applications Volume 009, Aticle ID , 5 pages doi:0.55/009/ Reseach Aticle Schu-Convexity fo a Class of Symmetic Functions and Its Applications Wei-Feng Xia and Yu-Ming Chu School of Teache Education, Huzhou Teaches College, Huzhou 33000, China Depatment of Mathematics, Huzhou Teaches College, Huzhou 33000, China Coespondence should be addessed to Yu-Ming Chu, chuyuming005@yahoo.com.cn Received 6 May 009; Accepted 4 Septembe 009 Recommended by Jozef Banas Fo x x,x,...,x n R n, the symmetic function φ n x, is defined by φ n x, φ n x, x,...,x n ; i <i <i n j x i j / /,whee,,...,nand i,i,...,i n ae positive integes. In this aticle, the Schu convexity, Schu multiplicative convexity and Schu hamonic convexity of φ n x, ae discussed. As applications, some inequalities ae established by use of the theoy of majoization. Copyight q 009 W.-F. Xia and Y.-M. Chu. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited.. Intoduction Thoughout this pape we use R n to denote the n-dimensional Euclidean space ove the field of eal numbes and R n {x,x,...,x n : x i > 0, i,,...,n}. In paticula, we use R to denote R. Fo the sake of convenience, we use the following notation system. Fo x x,x,...,x n,yy,y,...,y n R n,andα>0, let x y x y,x y,...,x n y n, xy x y,x y,...,x n y n, αx αx,αx,...,αx n, x α x α,xα,...,xα n, x,,..., x x x n,

2 Jounal of Inequalities and Applications log x log x, log x,...,log x n, e x e x,e x,...,e x n.. The notion of Schu convexity was fist intoduced by Schu in 93. Ithas many impotant applications in analytic inequalities 7, combinatoial optimization 8, isopeimetic poblem fo polytopes 9, linea egession 0, gaphs and matices, gamma and digamma functions, eliability and availability 3, and othe elated fields. The following definition fo Schu convex o concave can be found in, 3, 7 and the efeences theein. Definition.. Let E R n n be a set, a eal-valued function F on E is called a Schu convex function if Fx,x,...,x n F y,y,...,y n. fo each pai of n-tuples x x,...,x n and y y,...,y n on E, such that x is majoized by y in symbols x y, thatis, k x i i k y i, k,,...,n, i n n x i y i, i i.3 whee x i denotes the ith lagest component in x. F is called Schu concave if F is Schu convex. The notation of multiplicative convexity was fist intoduced by Montel 4. The Schu multiplicative convexity was investigated by Niculescu 5, Guan 7, andchuet al. 6. Definition. see 7, 6. LetE R n n be a set, a eal-valued function F : E R is called a Schu multiplicatively convex function on E if Fx,x,...,x n F y,y,...,y n.4 fo each pai of n-tuples x x,x,...,x n and y y,y,...,y n on E, such that x is logaithmically majoized by y in symbols log x log y, thatis, k x i i k y i, k,,...,n, i n n x i y i. i i.5 Howeve F is called Schu multiplicatively concave if /F is Schu multiplicatively convex.

3 Jounal of Inequalities and Applications 3 In pape 7, Andeson et al. discussed an attactive class of inequalities, which aise fom the notion of hamonic convex functions. Hee, we intoduce the notion of Schu hamonic convexity. Definition.3. Let E R n n be a set. A eal-valued function F on E is called a Schu hamonic convex function if Fx,x,...,x n F y,y,...,y n.6 fo each pai of n-tuples x x,x,...,x n and y y,y,...,y n on E, such that /x /y. F is called a Schu hamonic concave function on E if.6 is evesed. The main pupose of this pape is to discuss the Schu convexity, Schu multiplicative convexity, and Schu hamonic convexity of the following symmetic function: φ n x, φ n x,x,...,x n ; i <i <i n j /,.7 whee x x,x,...,x n R n n,,,...,n,andi,i,...,i ae positive integes. As applications, some inequalities ae established by use of the theoy of majoization.. Lemmas In ode to establish ou main esults we need seveal lemmas, which we pesent in this section. The following lemma is so-called Schu s condition which is vey useful fo detemining whethe o not a given function is Schu convex o Schu concave. Lemma. see 6, 7, 8. Let f : R n 0, n R 0, be a continuous symmetic function. If f is diffeentiable in R n,thenf is Schu convex if and only if f xi x j f 0 x i x j. fo all i, j,,...,n and x x,...,x n R n.alsof is Schu concave if and only if. is evesed fo all i, j,,...,n and x x,...,x n R n. Hee, f is a symmetic function in R n meaning that fpxfx fo any x R n and any n n pemutation matix P. Remak.. Since f is symmetic, the Schu s condition in Lemma., thatis,. can be educed to f x x f 0. x x.

4 4 Jounal of Inequalities and Applications Lemma.3 see 7, 6. Let f : R n R be a continuous symmtic function. If f is diffeentiable in R n,thenf is Schu multiplicatively convex if and only if f f log x log x x x 0 x x.3 fo all x x,x,...,x n R n.alsof is Schu multiplicatively concave if and only if.3 is evesed. Lemma.4. Let f : R n R be a continuous symmetic function. If f is diffeentiable in R n,then f is Schu hamonic convex if and only if x x x f x f 0 x x.4 fo all x x,x,...,x n R n.alsof is Schu hamonic concave if and only if.4 is evesed. Poof. Fom Definitions. and.3, we clealy see the fact that f : R n R is Schu hamonic convex if and only if Fx /f/x : R n R is Schu concave. This fact, Lemma. and Remak. togethe with elementay calculation imply that Lemma.4 is tue. Lemma.5 see 5, 6. Let x x,x,...,x n R n and n i x i s.ifc s, then c x c nc/s x nc/s, c x nc/s,..., c x n x,x,...,x n x. nc/s Lemma.6 see 6. Let x x,x,...,x n R n and n i x i s.ifc 0, then c x c nc/s x nc/s, c x nc/s,..., c x n x,x,...,x n x. nc/s.5.6 Lemma.7 see 9. Suppose that x x,x,...,x n R n and n i x i s. If 0 λ, then s λx s n λ λx n λ, s λx n λ,...,s λx n x,x,...,x n x..7 n λ 3. Main Results Theoem 3.. Fo {,,...,n}, the symmetic function φ n x, is Schu concave in R n. Poof. By Lemma. and Remak., we only need to pove that φn x, x x x φ nx, 0 x 3. fo all x x,x,...,x n R n and,,...,n.

5 Jounal of Inequalities and Applications 5 The poof is divided into fou cases. Case. If, then.7 leads to φ n x, φ n x,x,...,x n ; n x i. 3. x i i Howeve 3. and elementay computation lead to φn x, x x x φ nx, x x x x x x x x x φ nx, Case. If n and n, then.7 yields φ n x, n φ n x,x,...,x n ; n n i x i x i /n. 3.4 Fom 3.4 and elementay computation, we have φn x, n x x x φ nx, n x x x x n x n x x i x i x i /n Case 3. If n 3and, then by.7 we have φ n x, φ n x,x,,x n ; x x / n x x / j φ n x,x 3,...,x n ; x x x j3 x j x x / n x x / j φ n x,x 3,...,x n ;. x x x j3 x j 3.6

6 6 Jounal of Inequalities and Applications Elementay computation and 3.6 yield φn x, x x x φ nx, x x x x x φ n x, x x x x x x [ ] n x x 3 x x x j xj 0. x x j x x j x x j x x j j3 3.7 Case 4. If n 4and3 n, then fom.7, we have φ n x, φ n x,x,...,x n ; φ n x,x 3,...,x n ; 3 i <i < <i n φ n x,x 3,...,x n ; 3 i <i < <i n φn x, x x x 3 i <i < <i n x x x x 3 i <i < <i n x x x x φ nx, x x x j x x j j / j /, / / 3.8 x x x x x x 3 i <i < <i n x /x x /x j / 3 i <i < <i n x x x x j / x /x j / x /x j / φ nx, Theefoe, 3. follows fom Cases 4 and the poof of Theoem 3. is completed.

7 Jounal of Inequalities and Applications 7 Fo the Schu multiplicative convexity o concavity of φ n x,, wehavethefollowing theoem Theoem 3.. It holds that φ n x, is Schu multiplicatively concave in, n. Poof. Accoding to Lemma.3 we only need to pove that log x log x x φ n x, x φ n x, x 0 x 3.0 fo all x x,x,...,n, n and,,...,n.then poof is divided into fou cases. Case. If, then 3. leads to log x log x x φ n x, x φ n x, log x log x x x x φ n x, 0. x x x 3. Case. If n, n, then 3.4 yields log x log x x φ n x, n x log x log x x x n x x φ n x, n x x n x i x x x i i /n Case 3. If n 3and, then 3.6 implies φ n x, log x log x x x φ n x, x x φ nx, log x log x x x x x x x x / x x / x n x x x x x j / x j x / x x j / x j x / x x j / 0. x j j3 3.3

8 8 Jounal of Inequalities and Applications Case 4. If n 4and3 n, then fom 3.8 we have φ n x, log x log x x x log x log x x x x x 3 i <i < <i n 3 i <i < <i n φ n x, x x x x x / x x / x j / x x x x j / x /x j / x /x j / φ nx, Theefoe, Theoem 3. follows fom 3.0 and Cases 4. Remak 3.3. Fom 3. and 3. we know that φ n x, is Schu multiplicatively concave in 0, n and φ n x, n is Schu multiplicatively convex in 0, n. Theoem 3.4. Fo {,,...,n}, the symmetic function φ n x, is Schu hamonic convex in R n. Poof. Accoding to Lemma.4 we only need to pove that x x x φ n x, x φ n x, 0 x x 3.5 fo all x x,x,...,x n R n and,,...,n. The poof is divided into fou cases. Case. If, then fom 3. we have x x x φ n x, x x φ n x, x x x x x φ nx, Case. If n and n, then 3.4 leads to x x x φ n x, n x φ n x, n x x x x x x n /n x i 0. x x n x x x i i 3.7

9 Jounal of Inequalities and Applications 9 Case 3. If n 3and, then 3.6 yields x x x φ n x, x φ n x, x x x x φ n x, x x n x x x x j x x j 3x x x j xj x x j x x j x x j x x j j Case 4. If n 4and3 n, then 3.8 implies x x x φ n x, x φ n x, x x x x φ n x, x x x x x x 3 i <i < <i n x / x x / x j / 3 i <i < <i n x x x x x x j / x /x j / x / x j / Theefoe, 3.5 follows fom Cases 4 and the poof of Theoem 3.4 is completed. 4. Applications In this section, we establish some inequalities by use of Theoems 3., 3. and 3.4 and the theoy of majoization. Theoem 4.. If x x,x,,x n R n, s n i x i,h n x n/ n i /x i, and {,,...,n}, then φ n x, φ n c x/nc/s, fo c s; φ n x, φ n ch n x /cx x, fo c n i /x i ; 3 φ n x, φ n c x/nc/s, fo c 0; 4 φ n x, φ n ch n x/cx x, fo c 0; 5 φ n x, φ n s λx/n λ, fo 0 λ ;

10 0 Jounal of Inequalities and Applications 6 φ n x, φ n n λ/ n i /x i λ/x, fo 0 λ ; 7 φ n x, φ n s λx/n λ, fo 0 λ ; 8 φ n x, φ n n λ/ n i /x i λ/x, fo 0 λ. Poof. Theoem 4. follows fom Theoem 3., Theoem 3.4 and Lemmas.5.7 togethe with the fact that s λx s n λ λx n λ, s λx n λ,...,s λx n x,x,...,x n x. 4. n λ Theoem 4.. If x x,x,,x n R n, A n x /n n i x i, and {,,...,n}, then i ii i <i < <i n i <i < <i n j j / / [ ] A n x n!/!n! ; A n x [ ] n!/!n!. A n x 4. Poof. We clealy see that A n x,a n x,...,a n x x,x,...,x n x. 4.3 Theefoe, Theoem 4.i follows fom 4.3 and Theoem 3. togethe with.7,and Theoem 4.ii follows fom 4.3 and Theoem 3.4 togethe with.7. If we take intheoem 4.i and n in Theoem 4., espectively, then we have the following coollay. Coollay 4.3. If x x,x,...,x n R n and G n x n i x i /n,then i G n x G n x A nx A n x ; x ii A n A nx x A n x ; iii A n x A n x. 4.4 Remak 4.4. If we take n i x i in Coollay 4.3i, then we obtain the Weiestass inequality: see 0, page 60 n n n. x i i 4.5

11 Jounal of Inequalities and Applications Theoem 4.5. If x x,x,...,x n R n and {,,...,n}, then i ii i <i < <i n i <i < <i n j j / / [ ] H n x n!/!n! ; H n x [ ] n!/!n!. H n x 4.6 Poof. We clealy see that H n x, H n x,...,,,..., H n x x x x. x n 4.7 Theefoe, Theoem 4.5i follows fom 4.7 and Theoem 3.4 togethe with.7,and Theoem 4.5ii follows fom 4.7 and Theoem 3. togethe with.7. If we take and n in Theoem 4.5, espectively, then we get the following coollay. Coollay 4.6. If x x,x,...,x n R n,then i G n x G n x H nx H n x ; ii G n x H n x; x iii A n H nx x H n x ; iv A n x H n x. 4.8 Theoem 4.7. If x x,x,...,x n, n and {,,...,n}, then i <i < <i n j / [ ] G n x n!/!n!. 4.9 G n x Poof. We clealy see that logg n x,g n x,...,g n x logx,x,...,x n. 4.0 Theefoe, Theoem 4.7 follows fom 4.0, Theoem 3.,and.7.

12 Jounal of Inequalities and Applications If we take and n in Theoem 4.7, espectively, then we get the following coollay. Coollay 4.8. If x x,x,...,x n, n,then x i A n G nx x G n x ; ii G n x G n x. 4. Remak 4.9. Fom Remak 3.3 and 4.0 togethe with.7 we clealy see that inequality in Coollay 4.8i is evesed fo x 0, n and inequality in Coollay 4.8ii is tue fo x R n. Theoem 4.0. If x x,x,,x n R n, then i i <i <i n j / [ n i x i n i x ] n!/!n! n!/!n! i fo n ; ii iii n x i n i x i x i n i i <i <i n j i x i n ; / n i x i n!/!n! n!/!n! fo n ; iv n x i i n i x n i. 4. Poof. Theoem 4.0 follows fom Theoems 3., 3.4, and.7 togethe with the fact that n x, x,..., x n x i,,,...,. 4.3 i Theoem 4.. Let A A A A n be an n-dimensional simplex in R n and let P be an abitay point in the inteio of A. IfB i is the intesection point of staight line A i P and hypeplane i A A A i A i A n,i,,...,n, then fo {,,...,n } one has i i <i <i n j PB ij A ij B ij PB ij / n!/!n! ; n

13 Jounal of Inequalities and Applications 3 ii iii iv i <i <i n i <i <i n i <i <i n j j j A ij B ij A ij B ij PB ij PA ij A ij B ij PA ij A ij B ij A ij B ij PA ij / / / [ ] n n!/!n! ; n [ ] n n!/!n! ; n [ ] n n!/!n!. n 4.4 Poof. It is easy to see that n i PB i/a i B i and n i PA i/a i B i n, these identical equations imply n, n,..., PB, n A B n n, n n,..., n n PB A B,..., PA A B, PA A B,..., PB n A n B n PA n A n B n,. 4.5 Theefoe, Theoem 4. follows fom 4.5, Theoems 3., 3.4,and.7. Remak 4.. Mitinovic et al., pages established a seies of inequalities fo PA i /A i B i and PB i /A i B i, i,,...,n. Obvious, ou inequalities in Theoem 4. ae diffeent fom theis. Theoem 4.3. Suppose that A M n C n is a complex matix, λ,λ,...,λ n, and σ,σ,...,σ n ae the eigenvalues and singula values of A, espectively. If A is a positive definite Hemitian matix and {,,...,n}, then i ii iii iv i <i <i n i <i <i n i <i <i n i <i <i n j j λ ij λ ij λ ij / λ ij λ ij j j ta λ ij / / [ ] ta n!/!n! ; n ta [ ] n n!/!n! ; n ta [ ] n n!/!n! deti A n ; deti A / [ ] n!/!n! ta n ; deta

14 4 Jounal of Inequalities and Applications v i <i <i n λ ij n i λ i n i σ i j / i <i <i n σ ij n i λ i n i σ i j /. 4.6 Poof. i ii We clealy see that λ i > 0 i,,...,n and n i λ i ta, then we have ta n, ta n,...,ta n λ,λ,...,λ n. 4.7 Theefoe, Theoem 4.3i and ii follows fom 4.7, Theoems 3., 3.4,and.7. iii It is easy to see that λ, λ,..., λ n ae the eigenvalues of matix I A and n i λ ideti A, then we get n log deti A, deti n A,..., n deti A log λ, λ,..., λ n λ i, i,,...,n. 4.8 Theefoe, Theoem 4.3iii follows fom 4.8, Theoem 3.,and.7. iv It is not difficult to veify that ta log n, det A ta n det A,..., ta ta n log, ta,..., ta, det A λ λ λ n ta λ i, i,,...,n. 4.9 Theefoe, Theoem 4.3iv follows fom 4.9,andTheoem 3. togethe with.7. v A esult due to Weyl gives logλ,λ,...,λ n logσ,σ,...,σ n. 4.0 Fom 4.0, we clealy see that n i log λ i n i σ n i i, λ i n i σ i,..., λ λ n i log λ i n i σ i, σ n i λ i n i σ i λ i, n i λ i n λ n i σ i n i λ i n i σ n i i,..., λ i n σ σ n n i λ i n i σ i, i,,...,n. σ i i σ i, 4. Theefoe, Theoem 4.3v follows fom 4., Theoem 3.,and.7.

15 Jounal of Inequalities and Applications 5 Acknowledgment This wok was suppoted by the National Science Foundation of China no and the Natual Science Foundation of Zhejiang Povince no. D , Y6078, Y Refeences I. Schu, Übe eine klasse von mittelbildungen mit anwendungen auf die deteminantentheoie, Sitzungsbeichte de Beline Mathematischen Gesellschaft, vol., pp. 9 0, 93. G. H. Hady, J. E. Littlewood, and G. Pólya, Some simple inequalities satisfied by convex functions, Messenge of Mathematics, vol. 58, pp. 45 5, C. Stepolhkepniak, An effective chaacteization of Schu-convex functions with applications, Jounal of Convex Analysis, vol. 4, no., pp , J. S. Aujla and F. C. Silva, Weak majoization inequalities and convex functions, Linea Algeba and Its Applications, vol. 369, pp. 7 33, K. Guan, The Hamy symmetic function and its genealization, Mathematical Inequalities & Applications, vol. 9, no. 4, pp , K.-Z. Guan, Schu-convexity of the complete symmetic function, Mathematical Inequalities & Applications, vol. 9, no. 4, pp , K.-Z. Guan, Some popeties of a class of symmetic functions, Jounal of Mathematical Analysis and Applications, vol. 336, no., pp , F. K. Hwang and U. G. Rothblum, Patition-optimization with Schu convex sum objective functions, SIAM Jounal on Discete Mathematics, vol. 8, no. 3, pp. 5 54, X.-M. Zhang, Schu-convex functions and isopeimetic inequalities, Poceedings of the Ameican Mathematical Society, vol. 6, no., pp , C. Stepolhkepniak, Stochastic odeing and Schu-convex functions in compaison of linea expeiments, Metika, vol. 36, no. 5, pp. 9 98, 989. G. M. Constantine, Schu convex functions on the specta of gaphs, Discete Mathematics, vol. 45, no. -3, pp. 8 88, 983. M. Mekle, Convexity, Schu-convexity and bounds fo the gamma function involving the digamma function, The Rocky Mountain Jounal of Mathematics, vol. 8, no. 3, pp , F. K. Hwang, U. G. Rothblum, and L. Shepp, Monotone optimal multipatitions using Schu convexity with espect to patial odes, SIAM Jounal on Discete Mathematics, vol. 6, no. 4, pp , P. Montel, Su les fonctions convexes et les fonctions soushamoniques, Jounal de Mathématiques, vol. 7, no. 9, pp. 9 60, C. P. Niculescu, Convexity accoding to the geometic mean, Mathematical Inequalities & Applications, vol. 3, no., pp , Y. Chu, X. Zhang, and G. Wang, The Schu geometical convexity of the extended mean values, Jounal of Convex Analysis, vol. 5, no. 4, pp , G. D. Andeson, M. K. Vamanamuthy, and M. Vuoinen, Genealized convexity and inequalities, Jounal of Mathematical Analysis and Applications, vol. 335, no., pp , A. W. Mashall and I. Olkin, Inequalities: Theoy of Majoization and Its Applications, vol. 43 of Mathematics in Science and Engineeing, Academic Pess, New Yok, NY, USA, S.-H. Wu, Genealization and shapness of the powe means inequality and thei applications, Jounal of Mathematical Analysis and Applications, vol. 3, no., pp , P. S. Bullen, A Dictionay of Inequalities, vol. 97, Longman, Halow, UK, 998. D. S. Mitinović, J. E. Pečaić, and V. Volenec, Recent Advances in Geometic Inequalities, vol. 8, Kluwe Academic Publishes, Dodecht, The Nethelands, 989. H. Weyl, Inequalities between the two kinds of eigenvalues of a linea tansfomation, Poceedings of the National Academy of Sciences of the United States of Ameica, vol. 35, pp , 949.

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