Acta mathematica scientia, Series B >
THE SCHUR HARMONIC CONVEXITY FOR A CLASS OF SYMMETRIC FUNCTIONS
Received date: 2008-08-23
Online published: 2010-09-20
Supported by
This work was supported by NSFC (60850005) and NSF of Zhejiang Province (D7080080, Y7080185, Y607128).
In this article, we prove that the symmetric function
Fn*(x, r)=∑i1+i2+…+in=r(x1i1x2i2…xnin )1/r is Schur harmonic convex for x ∈R+n and r ∈ N={1, 2, 3, …}. As its applications, some analytic inequalities are established.
Key words： symmetric function; Schur convex; Schur harmonic convex
CHU Yu-Ming , SUN Tian-Chuan . THE SCHUR HARMONIC CONVEXITY FOR A CLASS OF SYMMETRIC FUNCTIONS[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1501 -1506 . DOI: 10.1016/S0252-9602(10)60142-7
[1] Guan K Z. The Hamy symmetric function and its generalization. Math Inequal Appl, 2006, 9(4): 797--805
[2] Guan K Z. A class of symmetric functions for multiplicatively convex function. Math Inequal Appl, 2007, 10(4): 745--753
[3] Aczél J. A generalization of the notion of convex function. Norske Vid Selsk Forh Trondhjem, 1947, 19(24): 87--90
[4] Vamanamurthy M K, Vuorinen M. Inequalities for means. J Math Anal Appl, 1994, 183(1): 155--166
[5] Roberts A W, Varberg D E. Convex Functions. New York: Academic Press, 1973
[6] Niculescu C P, Persson L E. Convex Functions and Their Applications. New York: Springer-Verlag, 2006
[7] Wu H X, Zou D X. Convex functions and their applications in probability theory. Acta Math Sci, 1990, 10(4): 365--373 (in Chinese)
[8] Cheng L X, Zhang F. Differentiability of convex functions and Asplund spaces. Acta Math Sci, 1995, 15(2): 171--179
[9] Ruan Y B, Chen S X. Approximation of convex functions on the dual spaces. Acta Math Sci, 2004, 24A(1): 116--122 (in Chinese)
[10] Anderson G D, Vamanamurthy M K, Vuorinen M. Generalized convexity and inequalities. J Math Anal Appl, 2007, 335(2): 1294--1308
[11] Marshall A W, Olkin I. Inequalities: Theory of Majorization and Its Applications. New York: Academic Press, 1979
[12] Hardy G H, Littleewood J E, P\'{o}lya G. Some simple inequalities satisfied by convex functions. Messenger Math, 1929, 58: 145--152
[13] Zhang X M. Schur-convex functions and isoperimetric inequalities. Proc Amer Math Soc, 1998, 126(2): 461--470
[14] Aujla J S, Silva F C. Weak majorization inequalities and convex functions. Linear Algebra Appl, 2003, 369: 217--233
[15] Qi F, Sándor J, Dragomir S S, Sofo A. Notes on the Schur-convexity of the extended mean values. Taiwanese J Math, 2005, 9(3): 411--420
[16] Chu Y M, Zhang X M. Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave. J Math Kyoto Univ, 2008, 48(1): 229--238
[17] Chu Y M, Zhang X M, Wang G D. The Schur geometrical convexity of the extended mean values. J Convex Anal, 2008, 15(4): 707--718
[18] Stepniak C. Stochastic ordering and Schur-convex functions in comparsion of linear experiments. Metrika, 1989, 36(5): 291--298
[19] Constantine G M. Schur convex functions on the spectra of graphs. Discrete Math, 1983, 45(2/3): 181--188
[20] Hwang F K, Rothblum U G. Partition-optimization with Schur convex sum objective functions. SIAM J Discrete Math, 2004/2005, 18(3): 512--524
[21] Forcina A, Giovagnoli A. Homogeneity indices and Schur-convex functions. Statistica (Bologna), 1982, 42(4): 529--542
[22] Merkle M. Convexity, Schur-convexity and bounds for the gamma function involving the digamma function. Rocky Mountain J Math, 1998, 28(3): 1053--1066
[23] Shaked M, Shanthikumar J G, Tong Y L. Parametric Schur convexity and arrangement monotonicity properties of partial sums. J Multivariate Anal, 1995, 53(2): 293--310
[24] Hwang F K, Rothblum U G, Shepp L. Monotone optimal multipartitions using Schur convexity with pespect to partial orders. SIAM J Discrete Math, 1993, 6(4): 533--547
[25] Guan K Z. Schur-convexity of the complete symmetric function. Math Inequal Appl, 2006, 9(4): 567--576
/
| 〈 |
|
〉 |