[1] Schur I. Uber¨ eine Klasse von Mittelbildungen mit Anwendungen auf die determinantentheorie. Sitzunsber
Berlin Math Ges, 1923, 22: 9–20

[2] Hardy G H, Littlewood J E, P′olya G. Some simple inequalities satisfied by convex function. Messenger
Math, 1929, 58: 145–152

[3] Zhang X M. Schur-convex functions and isoperimetric inequalities. Proc Amer Math Soc, 1998, 126(2): 461–470

[4] Marshall A W, Olkin I. Inequalities: Theory of Majorization and Its Applications. New York: Academic Press, 1979

[5] Jiang W D. Some properties of dual form of the Hamy’s symmetric function. J Math Inequal, 2007, 1(1):
117–125

[6] Shi H N, Jiang Y M, Jiang W D. Schur-convexity and Schur-geometrically concavity of Gini means. Comput Math Appl, 2009, 57(2): 266–274

[7] Shi H N. Schur-convex functions related to Hadamard-type inequalities. J Math Inequal, 2007, 1(1): 127–136

[8] Guan K Z. Some properties of a class of symmetric functions. J Math Anal Appl, 2007, 336(1): 70–80

[9] Guan K Z. The Hamy symmetric function and its generalization. Math Inequal Appl, 2006, 9(4): 797–805

[10] Guan K Z. Schur-convexity of the complete symmetric function. Math Inequal Appl, 2006, 9(4): 567–576

[11] Guan K Z, Shen J H. Schur-convexity for a class of symmetric function and its applications. Math Inequal
Appl, 2006, 9(2): 199–210

[12] Chu Y M, Xia W F, Zhao T H. Schur convexity for a class of symmetric functions. Sci China Math, 2010, 53(2): 465–474

[13] Shi H N, Wu S H, Qi F. An alternative note on the Schur-convexity of the extended mean values. Math
Inequal Appl, 2006, 9(2): 219–224

[14] Qi F. A note on Schur-convexity of extended mean values. Rocky Mountain J Math, 2005, 35(5): 1787–
1793

[15] Qi F, S′andor J, Dragomir S S, Sofo A. Notes on the Schur-convexity of the extended mean values. Tai-
wanese J Math, 2005, 9(3): 411–420

[16] Chu YM,Zhang XM.Necessary and su?cientconditions such that extended mean values areSchur-convex
or Schur-concave. J Math Kyoto Univ, 2008, 48(1): 229–238

[17] Chu Y M, Xia W F. Solution of an open problem for Schur convexity or concavity of the Gin mean values.
Sci China (Ser A), 2009, 52(10): 2099–2106

[18] Guan K Z. A class of symmetric functions for multiplicatively convex function. Math Inequal Appl, 2007,
10(4): 745–753

[19] 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

[20] Chu Y M, Wang G D, Zhang X H. The Schur multiplicative and harmonic convexities of the complete
symmetric function. Math Nachr, 2011, 284(5/6): 653–663

[21] Chu Y M, Sun T C. The Schur harmonic covexity for a class of symmetric functions. Acta Math Sci (Ser
B), 2010, 30(5): 1501–1506

[22] Wu S H. Generalization and sharpness of the power means inquality and their applications. J Math Anal
Appl, 2005, 312(2): 637–652