Journal of Inequalities and Applications

Journal of Inequalities and Applications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Schur-convexity of dual form of some symmetric functions Journal of Inequalities and Applications 2013, 2013:295 doi:10.1186/1029-242X-2013-295 Huan-Nan Shi (shihuannan@yahoo.com.cn) Jing Zhang (ldtzhangjing1@buu.edu.cn) ISSN Article type 1029-242X Research Submission date 16 April 2013 Acceptance date 8 June 2013 Publication date 17 June 2013 Article URL http://www.journalofinequalitiesandapplications.com/content/2013/1/ This peer-reviewed article can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2013 Shi and Zhang This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. JIA_477_edited [06/14 12:50] 1/8 SCHUR-CONVEXITY OF DUAL FORM OF SOME SYMMETRIC FUNCTIONS HUAN-NAN SHI AND JING ZHANG† Abstract. By the properties of a Schur-convex function, Schur-convexity of the dual form of some symmetric functions is simply proved. 2000 Mathematics Subject Classification: Primary 26D15; 05E05; 26B25. Keywords: majorization; Schur-convexity; inequality; symmetric functions; dual form; convex function 1. Introduction Throughout the article, R denotes the set of real numbers, x = (x1 , x2 , . . . , xn ) denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as Rn = {x = (x1 , . . . , xn ) : xi ∈ R, i = 1, . . . , n} , Rn+ = {x = (x1 , . . . , xn ) : xi > 0, i = 1, . . . , n}. In particular, the notations R and R+ denote R1 and R1+ , respectively. For convenience, we introduce some definitions as follows. Definition 1. [1, 2] Let x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) ∈ Rn . (i) x ≥ y means xi ≥ yi for all i = 1, 2, . . . , n. (ii) Let Ω ⊂ Rn , ϕ: Ω → R is said to be increasing if x ≥ y implies ϕ(x) ≥ ϕ(y). ϕ is said to be decreasing if and only if −ϕ is increasing. Definition 2. [1, 2] Let x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) ∈ Rn . k k (i) x is said to be majorized by y (in symbols x ≺ y) if i=1 x[i] ≤ i=1 y[i] n n for k = 1, 2, . . . , n − 1 and i=1 xi = i=1 yi , where x[1] ≥ · · · ≥ x[n] and y[1] ≥ · · · ≥ y[n] are rearrangements of x and y in a descending order. (ii) Let Ω ⊂ Rn , ϕ: Ω → R is said to be a Schur-convex function on Ω if x ≺ y on Ω implies ϕ (x) ≤ ϕ (y) . ϕ is said to be a Schur-concave function on Ω if and only if −ϕ is Schur-convex function on Ω. Definition 3. [1, 2] Let x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) ∈ Rn . (i) Ω ⊂ Rn is said to be a convex set if x, y ∈ Ω, 0 ≤ α ≤ 1 implies αx + (1 − α)y = (αx1 + (1 − α)y1 , . . . , αxn + (1 − α)yn ) ∈ Ω. (ii) Let Ω ⊂ Rn be a convex set. A function ϕ: Ω → R is said to be a convex function on Ω if ϕ (αx + (1 − α)y) ≤ αϕ(x) + (1 − α)ϕ(y) for all x, y ∈ Ω and all α ∈ [0, 1]. ϕ is said to be a concave function on Ω if and only if −ϕ is a convex function on Ω. † J. Zhang: Correspondence author. 1 JIA_477_edited [06/14 12:50] 2 2/8 H.-N. SHI AND J. ZHANG (iii) Let Ω ⊂ Rn . A function ϕ: Ω → R is said to be a log-convex function on Ω if the function ln ϕ is convex. Definition 4 ([1]). (i) Ω ⊂ Rn is called a symmetric set, if x ∈ Ω implies P x ∈ Ω for every n × n permutation matrix P . (ii) The function ϕ : Ω → R is called symmetric if for every permutation matrix P , ϕ(P x) = ϕ(x) for all x ∈ Ω. Theorem A. (Schur-convex function decision theorem)[1, p. 84]: Let Ω ⊂ Rn be symmetric and have a nonempty interior convex set. Ω0 is the interior of Ω. ϕ : Ω → R is continuous on Ω and differentiable in Ω0 . Then ϕ is the Schur − convex(Schur − concave)f unction if and only if ϕ is symmetric on Ω and   ∂ϕ ∂ϕ (x1 − x2 ) − ≥ 0(≤ 0) (1) ∂x1 ∂x2 holds for any x ∈ Ω0 . The Schur-convex functions were introduced by Schur in 1923 and have important applications in analytic inequalities, elementary quantum mechanics and quantum information theory. See [1]. In recent years, many scholars use the Schur-convex function decision theorem to determine the Schur-convexity of many symmetric functions. Wei-feng Xia et al.[3] proved that the symmetric function    k xij x = , k = 1, . . . , n, (2) Ek j=1 1+x 1 + xij 1≤i1 <...<ik ≤n Rn+ . is Schur-convex on Yu-ming Chu et al.[4] proved that the symmetric function    k xij x = Ek , k = 1, . . . , n, j=1 1 − xij 1−x (3) 1≤i1 <...<ik ≤n k−1 k−1 n is Schur-convex on [ 2(n−1) , 1)n and Schur-concave on [0, 2(n−1) ] . Wei-feng Xia and Yu-ming Chu [5] proved that the symmetric function    k 1 − xij 1−x = Ek , k = 1, . . . , n, j=1 x xij (4) 1≤i1 <...<ik ≤n 2n−k−1 n n is Schur-convex on (0, 2n−k−1 2(n−1) ] and Schur-concave on [ 2(n−1) , 1] . Wei-feng Xia and Yu-ming Chu [6] also proved that the symmetric function    k 1 + xij 1+x = Ek , k = 1, . . . , n, (5) j=1 1 − xij 1−x 1≤i1 <...<ik ≤n n is Schur-convex on (0, 1) . Hua Mei et al. [7] proved that the symmetric function     k  1 1 −x = Ek − xij , k = 1, . . . , n, j=1 x x ij (6) 1≤i1 <...<ik ≤n is Schur-convex on (0, 1)n . In this paper, by the properties of a Schur-convex function, we study Schurconvexity of the dual form of the above symmetric functions, and we obtained the following results. JIA_477_edited [06/14 12:50] 3/8 SCHUR-CONVEXITY OF DUAL FORM OF SOME SYMMETRIC FUNCTIONS 3 Theorem 1. The symmetric function    x ∗ = Ek 1+x k x ij , k = 1, . . . , n, 1 + xij (7) Theorem 2. The symmetric function    x = Ek∗ 1−x k x ij , k = 1, . . . , n, 1 − xij (8) k 1 − xij , k = 1, . . . , n, x ij (9) k 1 + xij , k = 1, . . . , n, 1 − xij (10) 1≤i1 <...<ik ≤n is a Schur-concave function on Rn+ . 1≤i1 <...<ik ≤n j=1 j=1 is a Schur-convex function on [ 21 , 1)n . Theorem 3. The symmetric function    1−x Ek∗ = x 1≤i1 <...<ik ≤n j=1 is a Schur-convex function on (0, 21 ]n . Theorem 4. The symmetric function    1+x Ek∗ = 1−x 1≤i1 <...<ik ≤n j=1 is a Schur-convex function on (0, 1)n . Theorem 5. The symmetric function     k  1 1 ∗ −x = − xij , k = 1, . . . , n, Ek j=1 x xij (11) 1≤i1 <...<ik ≤n  √ is a Schur-convex function on 0, 5−2 n . 2. Lemmas To prove the above three theorems, we need the following lemmas. Lemma 1. [1, p. 97],[2] If ϕ is symmetric and convex (concave) on a symmetric convex set Ω, then ϕ is Schur-convex (Schur-concave) on Ω. Lemma 2. [2, p. 64] Let Ω ⊂ Rn , ϕ: Ω → R+ . Then log ϕ is Schur-convex (Schur-concave) if and only if ϕ is Schur-convex (Schur-concave). Lemma 3. [1, p. 642],[2] Let Ω ⊂ Rn be an open convex set, ϕ : Ω → R. For x, y ∈ Ω, define one variable function g(t) = ϕ (tx + (1 − t)y) on the interval (0, 1). Then ϕ is convex (concave) on Ω if and only if g is convex (concave) on [0, 1] for all x, y ∈ Ω. Lemma 4. Let x = (x1 , . . . , xm ) and y = (y1 , . . . , ym ) ∈ Rm + . Then the function p(t) = log g(t) is concave on [0, 1], where g(t) = m  j=1 txj + (1 − t)yj . 1 + txj + (1 − t)yj JIA_477_edited [06/14 12:50] 4 4/8 H.-N. SHI AND J. ZHANG Proof. ′ ′ p (t) = g (t) , g(t) where ′ g (t) = m  xj − y j . (1 + txj + (1 − t)yj )2 j=1 ′′ ′ g (t)g(t) − (g (t))2 p (t) = , g 2 (t) ′′ where ′′ g (t) = − m  j=1 2(xj − yj )2 . (1 + txj + (1 − t)yj )3 Thus, ′′ ′ g (t)g(t) − (g (t))2 ⎞ ⎛ ⎞⎛ ⎞2 ⎛ m m m 2    2(x − y ) x − y tx + (1 − t)y j j j j j j ⎠−⎝ ⎠⎝ ⎠ = ⎝− 3 2 (1 + tx + (1 − t)y ) 1 + tx + (1 − t)y (1 + tx + (1 − t)y ) j j j j j j j=1 j=1 j=1 ≤ 0, ′′ and then p (t) ≤ 0, that is, p(t) is concave on [0, 1]. The proof of Lemma 4 is completed. Lemma 5. Let x = (x1 , . . . , xm ) and y = (y1 , . . . , ym ) ∈ function q(t) = log ψ(t) is convex on [0, 1], where ψ(t) = m  j=1 txj + (1 − t)yj . 1 − txj − (1 − t)yj Proof. ′ ′ q (t) = ψ (t) , ψ(t) where ′ ψ (t) = m  j=1 xj − y j . (1 − txj − (1 − t)yj )2 ′′ ′ ψ (t)ψ(t) − (ψ (t))2 q (t) = , ψ 2 (t) ′′ where ′′ ψ (t) = m  j=1 2(xj − yj )2 . (1 − txj − (1 − t)yj )3  1 m 2, 1 . Then the JIA_477_edited [06/14 12:50] 5/8 SCHUR-CONVEXITY OF DUAL FORM OF SOME SYMMETRIC FUNCTIONS 5 By the Cauchy inequality, we have ′′ ′ ψ (t)ψ(t) − (ψ (t))2 ⎞ ⎛ ⎞⎛ ⎞2 ⎛ m m m 2    x − y tx + (1 − t)y 2(x − y ) j j j j j j ⎠−⎝ ⎠⎝ ⎠ =⎝ 3 (1 − tx − (1 − t)y ) 1 − tx − (1 − t)y (1 − tx − (1 − t)yj )2 j j j j j j=1 j=1 j=1 ⎞2 ⎞2 ⎛  m  txj + (1 − t)yj x − y j j ⎠ ⎠ −⎝ ≥⎝ 3  (1 − txj − (1 − t)yj )2 (1 − txj − (1 − t)yj ) 2 1 − txj − (1 − t)yj j=1 j=1 ⎛ ⎞2 ⎛ ⎞2  m √ m   2|xj − yj | txj + (1 − t)yj x − y j j ⎠ −⎝ ⎠ . =⎝ 2 2 (1 − tx − (1 − t)y ) (1 − tx − (1 − t)y ) j j j j j=1 j=1 ⎛ √ m  2|xj − yj | √  ′′ From xj , yj ∈ [ 12 , 1) it follows that 2 txj + (1 − t)yj ≥ 1, hence ψ (t)ψ(t) − ′ ′′ (ψ (t))2 ≥ 0, and then q (t) ≥ 0, that is, q(t) is convex on [0, 1]. The proof of Lemma 5 is completed.  Lemma 6. Let x = (x1 , . . . , xm ) and y = (y1 , . . . , ym ) ∈ (0, 12 ]m . Then the function r(t) = log ϕ(t) is convex on [0, 1], where ϕ(t) = m  1 − txj − (1 − t)yj . txj + (1 − t)yj j=1 Proof. ′ ′ r (t) = where ′ ϕ (t) = − m  j=1 ′′ ′′ r (t) = where ′′ ϕ (t) = ϕ (t) , ϕ(t) xj − yj . (txj + (1 − t)yj )2 ′ ϕ (t)ϕ(t) − (ϕ (t))2 , ϕ2 (t) m  j=1 2(xj − yj )2 . (txj + (1 − t)yj )3 By the Cauchy inequality, we have ′′ ′ ϕ (t)ϕ(t) − (ϕ (t))2 ⎛ ⎞ ⎛ ⎞⎛ ⎞2 m m m 2    2(x − y ) x − y 1 − tx − (1 − t)y j j j j j j⎠ ⎠⎝ ⎠ − ⎝− =⎝ 3 (tx + (1 − t)y ) tx + (1 − t)y (tx + (1 − t)yj )2 j j j j j j=1 j=1 j=1 ⎞2 ⎛ ⎞2 m  1 − txj − (1 − t)yj x − y j j ⎠ −⎝ ⎠  ≥⎝ 3 2 (tx + (1 − t)y ) 2 tx + (1 − t)y (tx + (1 − t)y ) j j j j j j j=1 j=1 ⎞2 ⎛ ⎞2 ⎛  √ m m   2|xj − yj | 1 − txj − (1 − t)yj x − y j j ⎠ −⎝ ⎠ . =⎝ 2 (tx + (1 − t)y ) (tx + (1 − t)yj )2 j j j j=1 j=1 ⎛ m  √ 2|xj − yj |  JIA_477_edited [06/14 12:50] 6 6/8 H.-N. SHI AND J. ZHANG √  ′′ From xj , yj ∈ (0, 12 ] it follows that 2 1 − txj − (1 − t)yj ≥ 1, hence ϕ (t)ϕ(t) − ′ ′′ (ϕ (t))2 ≥ 0, and then r (t) ≥ 0, that is, r(t) is convex on [0, 1]. The proof of Lemma 6 is completed.  Lemma 7. Let x = (x1 , . . . , xm ) and y = (y1 , . . . , ym ) ∈ (0, 1)m . Then the function h(t) = log f (t) is convex on [0, 1], where f (t) = m  1 + txj + (1 − t)yj . 1 − txj − (1 − t)yj j=1 Proof. ′ f (t) , h (t) = f (t) ′ where ′ f (t) = m  2(xj − yj ) . (1 − txj − (1 − t)yj )2 j=1 ′′ h (t) = where ′′ f (t) = ′′ ′ f (t)f (t) − (f (t))2 , f 2 (t) m  j=1 4(xj − yj )2 . (1 − txj − (1 − t)yj )3 By the Cauchy inequality, we have ′′ ′ f (t)f (t) − (f (t))2 ⎛ ⎞⎛ ⎞2 ⎞ ⎛ m m m 2    4(x − y ) 1 + tx + (1 − t)y 2(x − y ) j j j j⎠ j j ⎠⎝ ⎠ =⎝ −⎝ 3 2 (1 − tx − (1 − t)y ) 1 − tx − (1 − t)y (1 − tx − (1 − t)y ) j j j j j j j=1 j=1 j=1 ⎞2 ⎛ ⎞2 m  1 + txj + (1 − t)yj 2(xj − yj ) 2|xj − yj | ⎠ −⎝ ⎠ ≥⎝ 3  2 (1 − tx 2 1 − tx − (1 − t)y (1 − tx − (1 − t)y ) j − (1 − t)yj ) j j j j j=1 j=1 ⎞2 ⎛ ⎞2 ⎛  m m   2|xj − yj | 1 + txj + (1 − t)yj 2(x − y ) j j ⎠ −⎝ ⎠ . =⎝ 2 2 (1 − tx − (1 − t)y ) (1 − tx − (1 − t)y ) j j j j j=1 j=1 ⎛ m   √  ′′ From xj , yj ∈ (0, 1) it follows that 2 1 + txj + (1 − t)yj ≥ 1, hence f (t)f (t) − ′ ′′ (f (t))2 ≥ 0, and then h (t) ≥ 0, that is, h(t) is convex on [0, 1]. The proof of Lemma 7 is completed.   √ m 5 − 2 . Then Lemma 8. Let x = (x1 , . . . , xm ) and y = (y1 , . . . , ym ) ∈ 0, the function s(t) = log w(t) is convex on [0, 1], where  m   1 w(t) = − (txj + (1 − t)yj ) . txj + (1 − t)yj j=1 Proof. ′ w (t) , s (t) = w(t) ′ JIA_477_edited [06/14 12:50] 7/8 SCHUR-CONVEXITY OF DUAL FORM OF SOME SYMMETRIC FUNCTIONS where ′ w (t) = −  m  (xj − yj ) j=1 7  1 + 1 . (txj + (1 − t)yj )2 ′′ ′ w (t)w(t) − (w (t))2 , s (t) = w2 (t) ′′ where ′′ w (t) = m  j=1 2(xj − yj )2 . (txj + (1 − t)yj )3 By the Cauchy inequality, we have ′′ ′ w (t)w(t) − (w (t))2 ⎛ ⎞⎛ ⎞  m m  2   2(xj − yj ) 1 ⎠⎝ =⎝ − (txj + (1 − t)yj ) ⎠ 3 (tx + (1 − t)y ) tx + (1 − t)yj j j j j=1 j=1 ⎛ − ⎝− ⎛ ≥⎝ ⎛ m  j=1 m  j=1 (xj − yj ) √  ⎞  2 1 +1 ⎠ (txj + (1 − t)yj )2 2|xj − yj | 3 (txj + (1 − t)yj ) 2  m  ⎝ (xj − yj ) − j=1  ⎞2 1 − (txj + (1 − t)yj )⎠ txj + (1 − t)yj ⎞  2 1 +1 ⎠ (txj + (1 − t)yj )2 ⎞2 ⎛ ⎞2  m 2  2|xj − yj | 1 − (txj + (1 − t)yj )2 ⎠ − ⎝ (xj − yj ) 1 + (txj + (1 − t)yj ) ⎠ . =⎝ 2 (tx + (1 − t)y ) (txj + (1 − t)yj )2 j j j=1 j=1  √ √ 5 − 2 it follows that u2j ≤ 5−2. Let uj := txj +(1−t)yj . From xj , yj ∈ 0, Since √ u2j ≤ 5 − 2 ⇔ (u2j + 2)2 ≤ 5 ⇔ u4j + 4u2j − 1 ≤ 0 √  ⇔ 2(1 − u2j ) ≥ (1 + u2j )2 ⇔ 2 1 − u2j ≥ 1 + u2j , ⎛ m  ′′ √ ′ ′′ so w (t)w(t) − (w (t))2 ≥ 0, and then s (t) ≥ 0, that is, s(t) is convex on [0, 1]. The proof of Lemma 8 is completed.  3. Proof of main results Proof of Theorem 4: For any 1 ≤ i1 < · · · < ik ≤ n, by Lemma 3 and Lemma k 1+xi k 1+xi 7, it follows that log j=1 1−xij is convex on (0, 1)k . Obviously, log j=1 1−xij j j   k 1+xi 1+x = 1≤i1 <...<ik ≤n log j=1 1−xij is also convex on (0, 1)n , and then log Ek∗ 1−x j  1+x is convex on (0, 1)n . Furthermore, it is clear that log Ek∗ 1−x is symmetric on  1+x (0, 1)n . By Lemma 1, it follows that log Ek∗ 1−x is Schur-convex on (0, 1)n , and  1+x is also Schur-convex on (0, 1)n . then from Lemma 2 we conclude that Ek∗ 1−x JIA_477_edited 8 [06/14 12:50] 8/8 H.-N. SHI AND J. ZHANG The proof of Theorem 4 is completed. Similar to the proof of Theorem 4, we can use Lemma 4, Lemma 5, Lemma 6 and Lemma 8 respectively to prove Theorem 1, Theorem 2, Theorem 3 and Theorem 5; therefore we omit the details of the proof. Remark 1. Using the Schur-convex function decision theorem, Qinghua Liu et al.[8] have proved Theorem 3. Weifeng Xia and Yuming Chu [9] have proved that the symmetric function    k 1 + xij 1+x ∗ Ek = , k = 1, . . . , n, (12) j=1 x x ij 1≤i1 <...<ik ≤n is a Schur-convex function on Rn+ . The reader may wish to prove the inequality (12) by the properties of a Schurconvex function. Acknowledgments The work was supported by Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407). Thanks for the help. This article was typeset by using AMS-LATEX. References [1] A. W.Marshall, I.Olkin and B.C.Arnold, Inequalities: theory of majorization and its application (Second Edition). New York : Springer Press, 2011 [2] B.Y. Wang, Foundations of majorization inequalities, Beijing Normal Univ. Press, Beijing, China, 1990 (in Chinese). [3] W.F Xia, G.D. Wang and Y.M. Chu. 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Schur convexity with respect to a class of symmetric functions and their applications. Bulletin of Mathematical Analysis and Applications, 3 (3), 84-96 (2011), URL: http://www.bmathaa.org [10] H.-N.Shi, Theory of majorization and analytic inequalities, Harbin Institute of Technology Press,Harbin, China, 2012 (in Chinese). (H.-N. Shi) Department of Electronic Information, Teacher’s College, Beijing Union University, Beijing City, 100011, P.R.China E-mail address: shihuannan@yahoo.com.cn, sfthuannan@buu.edu.cn (J. Zhang) Basic Courses Department of Beijing Union University,Beijing 100101, P.R.China E-mail address: ldtzhangjing1@buu.edu.cn