SCHUR m-power CONVEXITY OF GEOMETRIC BONFERRONI MEAN

Transcription

1 ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N ( SCHUR m-power CONVEXITY OF GEOMETRIC BONFERRONI MEAN Huan-Nan Shi Deartment of Mathematics Longyan University Longyan Fujian Peole s Reublic of China and Deartment of Electronic Information Teacher s College Beijing Union University Beijing 000 Peole s Reublic of China sfthuannan@buu.edu.cn Shan-He Wu Deartment of Mathematics Longyan University Longyan Fujian Peole s Reublic of China shanhewu@gmail.com Abstract. In this aer the Schur m-ower convexity of the geometric Bonferroni mean for n variables is discussed. Keywords: Schur m-ower convexity, geometric Bonferroni means, majorization.. Introduction Throughout the aer R denotes the set of real numbers, x = (x, x 2,..., x n denotes n-tule (n-dimensional real vectors, the set of vectors can be written as R n = {x = (x, x 2,..., x n : x i R, i =, 2,..., n}, R n + = {x = (x, x 2,..., x n : x i 0, i =, 2,..., n}, R n ++ = {x = (x, x 2,..., x n : x i > 0, i =, 2,..., n}. In articular, the notations R, R + and R ++ denote R, R + and R ++, resectively. The Bonferroni mean has imortant alication in multi criteria decisionmaking (see2 7], it was initially roosed by Bonferroni ], as follows. Corresonding author

2 HUAN-NAN SHI, SHAN-HE WU 770 Definition. Let x = (x, x 2,..., x n R n + and, q 0, + q 0. Bonferroni mean is defined by The ( B,q (x = n(n i,j=,i j x i xq j Motivated by the Bonferroni mean B,q (x and the geometric mean G(x = n i= (x i n, Xia et al. 4] introduced a new mean which is called the geometric Bonferroni mean, i.e., Definition 2. Let x = (x, x 2,..., x n R n ++ and (, q R The geometric Bonferroni mean is defined by (2 GB,q (x = n (x i + qx j n(n. + q i,j=,i j Obviously, the geometric Bonferroni mean has the following roerties: (i GB,q (0, 0,..., 0 = 0. (ii GB,q (x, x,..., x = x, if x i = x for i =, 2,..., n. (iii GB,q (x GB,q (y, if x i y i for i =, 2,..., n. +q (iv min{x, x 2,..., x n } GB,q (x max{x, x 2,..., x n }. Furthermore, if q = 0, then the geometric Bonferroni mean reduces to the geometric mean, i.e., GB,0 (x = n n (x i n(n = (x i n = G(x. i,j=,i j In recent years, the theory of majorization has been used as an imortant tool in studying the roerties of the mean. Yang 8],9],0] generalized the notion of Schur convexity to Schur f-convexity and discussed the Schur m- ower convexity of Stolarsky means 8], Gini means 9] and Daróczy means 0]. Subsequently, the Schur m-ower convexity has evoked the interest of many researchers (see ], 2], 3], 4]. In this aer, we discuss the Schur m-ower convexity of the geometric Bonferroni mean GB,q (x, Our main results are stated in the following theorem. Theorem. Let x = (x, x 2,..., x n R n ++. For fixed (, q R 2 ++ and n 3, i= (i if m < 0, then GB,q (x is Schur m-ower convex; (ii if m = 0, then GB,q (x is Schur m-ower convex; (iii if m =, then GB,q (x is Schur m-ower concave; (iv if m 2, then GB,q (x is Schur m-ower concave..

3 SCHUR m-power CONVEXITY OF GEOMETRIC BONFERRONI MEAN Preliminaries We begin with recalling some basic concets and notations in the theory of majorization. For more details, we refer the reader to 5, 6]. Definition 3. Let x = (x, x 2,..., x n and y = (y, y 2,..., y n R n. (i x y means x i y i for all i =, 2,..., n. (ii Let Ω R n, ϕ: Ω R is said to be increasing if x y imlies ϕ(x ϕ(y. ϕ is said to be decreasing if and only if ϕ is increasing. Definition 4. Let x = (x, x 2,..., x n and y = (y, y 2,..., y n R n. (i x is said to be majorized by y (in symbols x y if k i= x i] k i= y i] for k =, 2,..., n and n i= x i = n i= y i, where x ] x 2] x n] and y ] y 2] y n] are rearrangements of x and y in a descending order. (ii Let Ω R n,the function ϕ: Ω R is said to be Schur convex on Ω if x y on Ω imlies ϕ (x ϕ (y. ϕ is said to be Schur concave function on Ω if and only if ϕ is Schur convex function on Ω. Definition 5. Let x = (x, x 2,..., x n and y = (y, y 2,..., y n R n. (i Ω R n is said to be a convex set if x, y Ω, 0 α imlies αx+( αy= (αx +( αy, αx 2 +( αy 2,..., αx n + ( αy n Ω. (ii Let Ω R n be convex set. A function ϕ: Ω R is said to be convex on Ω if ϕ (αx + ( αy αϕ(x + ( αϕ(y for all x, y Ω, and all α 0, ]. The function ϕ is said to be concave on Ω if and only if ϕ is convex function on Ω. Definition 6. (i A set Ω R n is called symmetric, if x Ω imlies xp Ω for every n n ermutation matrix P. (ii A function ϕ : Ω R is called symmetric if for every ermutation matrix P, ϕ(xp = ϕ(x for all x Ω. The first systematical study of the functions reserving the ordering of majorization was made by I. Schur in 923. In Schur s honor, such functions are said to be Schur convex (see 5]. It can be used extensively in analytic inequalities, combinatorial otimization, quantum hysics, information theory, and other related fields. The following roosition is called Schur s condition (see5]. It rovides an aroach for testing whether a vector valued function is Schur convex or not.

4 HUAN-NAN SHI, SHAN-HE WU 772 Proosition. Let Ω R n be symmetric and have a nonemty interior convex set. Ω 0 is the interior of Ω. ϕ : Ω R is continuous on Ω and differentiable in Ω 0. Then ϕ is the Schur convex function (Schur concave function if and only if ϕ is symmetric on Ω and ϕ(x (3 (x x 2 ϕ(x ] 0 ( 0 x x 2 holds for any x Ω 0. A generalization of Schur convex functions was introduced by Yang 8], as follows Definition 7. Let f : R ++ R be defined by x m (4 f(x = m, m 0; ln x, m = 0. Then a function ϕ : Ω R n ++ R is said to be Schur m-ower convex on Ω if (f(x, f(x 2,..., f(x n (f(y, f(y 2,..., f(y n for all (x, x 2,..., x n Ω and (y, y 2,..., y n Ω imlies φ(x φ(y. If ϕ is Schur m-ower convex, then we say that ϕ is Schur m-ower concave. Similarly to the Schur s condition mentioned above, Yang 8] gave a method of determining the Schur m-ower convex functions, i.e., Proosition 2. Let Ω R n ++ be a symmetric set with nonemty interior Ω and ϕ : Ω R be continuous on Ω and differentiable in Ω. Then ϕ is Schur m-ower convex (Schur m-ower concave on Ω if and only if ϕ is symmetric on Ω and (5 and x m xm 2 m x m ϕ(x x x m 2 ] ϕ(x 0 ( 0, if m 0 x 2 ] ϕ(x ϕ(x (6 (log x log x 2 x x 2 0 ( 0, if m = 0 x x 2 for all x Ω.

5 SCHUR m-power CONVEXITY OF GEOMETRIC BONFERRONI MEAN Proof of Theorem Proof. Note that the geometric Bonferroni mean is defined by where Q = GB,q (x = + q Taking the natural logarithm gives log GB,q (x = log n i,j=,i j log(x + qx j + log(x 2 + qx j ] + + log(x + qx 2 + log(x 2 + qx + (x i + qx j n(n. + q + n(n Q log(x i + qx + log(x i + qx 2 ] i,,i j log(x i + qx j. Hence, we have GB,q (x x GB,q (x x 2 = GB,q (x + n(n x + qx j q + + x + qx 2 x 2 + qx = GB,q (x + n(n x 2 + qx j q + + x + qx 2 x 2 + qx q x i + qx q x i + qx 2 It is easy to see that GB,q (x is symmetric on R n +. Without loss of generality, we may assume that x x 2. Direct comutation gives : = xm xm 2 m ( x m = (xm xm 2 GB,q (x mn(n GB,q (x x m 2 x x m ( x + qx j GB,q (x a 2 x m 2 x 2 + qx j

6 HUAN-NAN SHI, SHAN-HE WU q x m ( x m 2 x i + qx x i + qx 2 + x m qx m 2 x + qx 2 = (xm xm 2 GB,q (x mn(n + q + qx m x2 m ] x 2 + qx qx x 2 (x m x m 2 + x i (x m x m 2 (x i + qx (x i + qx 2 x x 2 (x m x m 2 + qx j (x m x m 2 (x + qx j (x 2 + qx j + x x 2 ( 2 + q 2 (x m x m 2 + 2q(x 2 m x 2 m (x + qx 2 (x 2 + qx 2 ]. If m < 0, then x m xm 2 0, x m x m 2 0, x m x m 2 0 and x 2 m x 2 m 2 0. Thus, 0. From Proosition 2, it follows that GB,q (x is Schur m-ower convex for x R n ++. If m 2, then x m xm 2 0, x m x m 2 0, x m x m 2 0 and x 2 m x 2 m 2 0. Thus, 0. By Proosition 2, we conclude that GB,q (x is Schur m-ower concave for x R n ++. If m =, then = (x x 2 2 GB,q (x 2 n(n (x + qx j (x 2 + qx j q 2 + (x i + qx (x i + qx 2 + ( q 2 ] (x + qx 2 (x 2 + qx 0. By using Proosition 2, we deduce that GB,q (x is Schur m-ower concave for x R n ++. If m = 0, then ( GB,q (x = (log x log x 2 x x = (log x log x 2 GB,q (x n(n + q GB,q (x x 2 x 2 x ( x + qx j x 2 x 2 + qx j x x 2 ( + x qx 2 + qx x ] 2 x i + qx x i + qx 2 x + qx 2 x 2 + qx

7 SCHUR m-power CONVEXITY OF GEOMETRIC BONFERRONI MEAN 775 = (x x 2 (log x log x 2 GB,q (x n(n + q 0. qx j (x + qx j (x 2 + qx j x i (x i + qx (x i + qx 2 + 2q(x + x 2 ] (x + qx 2 (x 2 + qx From Proosition 2, we conclude that GB,q (x is Schur m-ower convex for x R n ++. The roof of Theorem is comleted. Authors contributions. All authors contributed equally and significantly in this aer. All authors read and aroved the final manuscrit. Acknowledgements. This research was suorted by the Natural Science Foundation of Fujian rovince of China under Grant No.206J0023. References ] C. Bonferroni, Sulle medie multile di otenze, Bollettino dell Unione Matematica Italiana, 5 (950, ] R. R. Yager,On generalized Bonferroni mean oerators for multi-criteria aggregation, International Journal of Aroximate Reasoning, 50 (2009, ] J. H. Park, E. J. Park, Generalized fuzzy Bonferroni harmonic mean oerators and their alications in grou decision making, Journal of Alied Mathematics, 203 (203, Article ID , 4 ages. 4] M. Xia, Z. Xu, B. Zhu, Generalized intuitionistic fuzzy Bonferroni means, International Journal of Intelligent Systems, 27 (202, ] J. H. Park, J. Y. Kim,Intuitionistic fuzzy otimized weighted geometric Bonferroni means and their alications in Grou Decision Making, Fundamenta Informaticae, 44 (206, ] G. Beliakov, S. James, J. Mordelová, T. Rückschlossová, R. R. Yager, Generalized Bonferroni mean oerators in multicriteria aggregation, Fuzzy Sets and Systems, 6 (200, ] Z. Xu, R. R. Yager, Intuitionistic Fuzzy Bonferroni Means, IEEE Transactions on Systems, Man, and Cybernetics, Part B, 4 (20, ] Z. H. Yang, Schur ower convexity of Stolarsky means, Publ. Math. Debrecen, 80 (202, ] Z. H. Yang, Schur ower convexity of Gini means, Bull. Korean Math. Soc., 50 (203,

8 HUAN-NAN SHI, SHAN-HE WU 776 0] Z. H. Yang, Schur ower comvexity of the daróczy means, Math. Inequal. Al., 6 (203, ] Y. P. Deng, S. H. Wu, Y. M. Chu, D. He, The Schur convexity of the generalized Muirhead-Heronian means, Abstract and Alied Analysis, 204 (204, Article ID 70658, ages. 2] W. Wang, S. G. Yang, Schur m-ower convexity of a class of multilicatively convex functions and alications, Abstract and Alied Analysis, 204 (204, Article ID 25808, 2 ages. 3] Q. Xu, Reserch on Schur -ower convexity of the quotient of arithmetic mean and geometric mean, Journal of Fudan University(Natural Science, 54 (205, ] H. P. Yin, H. N. Shi, F. Qi, On Schur m-ower convexity for ratios of some means, J. Math. Inequal., 9 (205, ] A. W. Marsshall, I. Olkin, B. C. Arnold, Inequalities: Theory of majorization and its alication (Second Edition, Sringer, New York, 20. 6] B. Y. Wang, Foundations of majorization inequalities (Chinese, Beijing Normal University Press, Beijing, China, 990. Acceted: