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International Journal of Functional Analysis, Operator Theory and Applications

International Journal of Functional Analysis, Operator Theory and Applications
Volume 1, Issue 1, Pages 1 - 8 (June 2009)

THE SCHUR MULTIPLICATIVE CONVEXITY OF THE GENERALIZED MUIRHEAD MEAN VALUES Wei-Feng Xia (P. R. China) and Yu-Ming Chu (P. R. China) Received November 13, 2008 References:
[1] J. S. Aujla and F. C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003), 217-233. [2] J. L. Brenner and B. C. Carlson, Homogeneous mean values: weights and asymptotics, J. Math. Anal. Appl. 123(1) (1987), 265-280. [3] P. S. Bullen, D. S. Mitrinović and P. M. Vasić, Means and their Inequalities, D. Reidel Publishing Co., Dordrecht, 1988. [4] G. M. Constantine, Schur convex functions on the spectra of graphs, Discrete Math. 45(2-3) (1983), 181-188. [5] S. J. Dilworth, R. Howard and J. W. Roberts, A general theory of almost convex functions, Trans. Amer. Math. Soc. 358(8) (2006), 3413-3445. [6] N. Garofalo and F. Tournier, New properties of convex functions in the Heisenberg group, Trans. Amer. Math. Soc. 358(5) (2006), 2011-2055. [7] R. E. Greene and H. Wu, convex functions and manifolds of positive curvature, Acta Math. 137(1) (1976), 209-245. [8] R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: topological structure, Invent. Math. 63(1) (1981), 129-157. [9] F. K. Hwang and U. G. Rothblum, Partition-optimization with Schur convex sum objective functions, SIAM J. Discrete Math. 18(3) (2004/05), 512-524. [10] G. Z. Lu, J. J. Manfredi and B. Stroffolini, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19(1) (2004), 1-22. [11] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. [12] P. Montel, Sur les fonctions convexes et les fonctions sousharmoniques, J. Math. Pures Appl. (9) 7 (1928), 29-60. [13] C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl. 3(2) (2000), 155-167. [14] O. C. Schnürer, Convex functions with unbounded gradient, Results Math. 48 (2005), 158-161. [15] C. Stępniak, Stochastic ordering and Schur-convex functions in comparison of linear experiments, Metrika 36(1) (1989), 291-298. [16] V. Titarenko and A. Yagola, Linear ill-posed problems on sets of convex functions on two-dimensional sets, J. Inverse Ill-Posed Probl. 14(7) (2006), 735-750. [17] T. Trif, Monotonicity, comparison and Minkowski’s inequality for generalized Muirhead means in two variables, Mathematica 48(71) (1) (2006), 99-110. [18] X.-M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126(2) (1998), 461-470.

Keywords and phrases: generalized Muirhead mean values, Schur convex, Schur multiplicatively convex.

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