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Wang, Wen; Yang, Shiguo

Schur \(m\)-power convexity of a class of multiplicatively convex functions and applications. (English) Zbl 1470.26019

Abstr. Appl. Anal. 2014, Article ID 258108, 12 p. (2014).
Summary: We investigate the conditions under which the symmetric functions \(F_{n, k}(\mathbf{x}, r) = \prod_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} f \left((\sum_{j = 1}^k x_{i_j}^r)^{1 / r}\right)\), \(k = 1,2, \dots, n\), are Schur \(m\)-power convex for \(x \in \mathbb{R}_{+ +}^n\) and \(r > 0\). As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving the \(p\)th power mean and the arithmetic, the geometric, or the harmonic means are presented.

Cited in 6 Documents

MSC:

26B25 Convexity of real functions of several variables, generalizations
05E05 Symmetric functions and generalizations
26D15 Inequalities for sums, series and integrals
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[1] Bullen, P. S.; Mitrinovic, D. S.; Vasic, M., Handbook of Means and Theirs Inequality (2003), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 1035.26024
[2] Hara, T.; Uchiyama, M.; Takahasi, S.-E., A refinement of various mean inequalities, Journal of Inequalities and Applications, 2, 4, 387-395 (1998) · Zbl 0917.26017 · doi:10.1155/S1025583498000253
[3] Guan, K. Z., A class of symmetric functions for multiplicatively convex function, Mathematical Inequalities & Applications, 10, 4, 745-753 (2007) · Zbl 1144.26015 · doi:10.7153/mia-10-69
[4] Rovenţa, I., Schur convexity of a class of symmetric functions, Annals of the University of Craiova. Mathematics and Computer Science Series, 37, 1, 12-18 (2010) · Zbl 1224.26056
[5] Meng, J. X.; Chu, Y. M.; Tang, X. M., The Schur-harmonic-convexity of dual form of the Hamy symmetric function, Matematichki Vesnik, 62, 1, 37-46 (2010) · Zbl 1265.26040
[6] Shi, H. N.; Zhang, J., Schur-convexity, Schur-geometric and Harmonic convexity of dual form of a class symmetric functions, Journal of Inequalities and Applications, 2013, 295 (2013) · Zbl 1282.26039 · doi:10.1186/1029-242X-2013-295
[7] Yang, Zh.-H., Schur power convexity of Stolarsky means, Publicationes Mathematicae Debrecen, 80, 1-2, 43-66 (2012) · Zbl 1299.26035
[8] Yang, Zh.-H., Schur power convexity of Gini means, Bulletin of the Korean Mathematical Society, 50, 2, 485-498 (2013) · Zbl 1284.26016 · doi:10.4134/BKMS.2013.50.2.485
[9] Yang, Zh.-H., Schur power convexity of the Daróczy means, Mathematical Inequalities & Applications, 16, 3, 751-762 (2013) · Zbl 1277.26050 · doi:10.7153/mia-16-57
[10] Wang, W.; Yang, S. G., Schur m-power convexity of generalized Hamy symmetric function · Zbl 1332.05144
[11] Wang, W.; Yang, S. G., On the Schur m-power convexity for a class of symmetric functions · Zbl 1313.26019
[12] Wang, B. Y., Foundations of Majorization Inequalities (in Chinese) (1990), Beijing, China: Beijing Normal University Press, Beijing, China
[13] Marshall, A. W.; Olkin, I.; Arnold, B. C., Inequalities: Theory of Majorization and Its Applications (2011), New York, NY, USA: Springer, New York, NY, USA · Zbl 1219.26003 · doi:10.1007/978-0-387-68276-1
[14] Zhang, X. M., Geometrically-Convex Functions (in Chinese) (2004), Hefei, China: Anhui University Press, Hefei, China
[15] Chu, Y. M.; Sun, T. C., The Schur harmonic convexity for a class of symmetric functions, Acta Mathematica Scientia B, 30, 5, 1501-1506 (2010) · Zbl 1240.26027 · doi:10.1016/S0252-9602(10)60142-7
[16] Wu, Y.; Qi, F., Schur-harmonic convexity for differences of some means, Analysis, 32, 4, 263-270 (2012) · Zbl 1256.26010 · doi:10.1524/anly.2012.1171
[17] Niculescu, C. P., Convexity according to the geometric mean, Mathematical Inequalities & Applications, 3, 2, 155-167 (2000) · Zbl 0952.26006 · doi:10.7153/mia-03-19
[18] Long, B. Y.; Chu, Y. M., The Schur convexity and inequalities for a class of symmetric functions, Acta Mathematica Scientia A, 32, 1, 80-89 (2012) · Zbl 1265.26039
[19] Wu, S. H., Generalization and sharpness of the power means inequality and their applications, Journal of Mathematical Analysis and Applications, 312, 2, 637-652 (2005) · Zbl 1083.26018 · doi:10.1016/j.jmaa.2005.03.050
[20] Xia, W. F.; Chu, Y. M., Schur convexity with respect to a class of symmetric functions and their applications, Bulletin of Mathematical Analysis and Applications, 3, 3, 84-96 (2011) · Zbl 1314.05216
[21] Ku, H.-T.; Ku, M.-C.; Zhang, X.-M., Inequalities for symmetric means, symmetric harmonic means, and their applications, Bulletin of the Australian Mathematical Society, 56, 3, 409-420 (1997) · Zbl 0894.26010 · doi:10.1017/S0004972700031191
[22] Guan, K. Z., The Hamy symmetric function and its generalization, Mathematical Inequalities & Applications, 9, 4, 797-805 (2006) · Zbl 1107.05094 · doi:10.7153/mia-09-70
[23] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Some simple inequalities satisfied by convex functions, Messenger of Mathematics, 58, 145-152 (1929) · JFM 55.0740.04
[24] Chu, Y. M.; Xia, W. F.; Zhao, T. H., Schur convexity for a class of symmetric functions, Science China. Mathematics, 53, 2, 465-474 (2010) · Zbl 1230.05278 · doi:10.1007/s11425-009-0188-2
[25] Qi, F.; Sándor, J.; Dragomir, S. S.; Sofo, A., Notes on the Schur-convexity of the extended mean values, Taiwanese Journal of Mathematics, 9, 3, 411-420 (2005) · Zbl 1086.26008
[26] Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M., Generalized convexity and inequalities, Journal of Mathematical Analysis and Applications, 335, 2, 1294-1308 (2007) · Zbl 1125.26017 · doi:10.1016/j.jmaa.2007.02.016
[27] Li, D.-M.; Shi, H.-N., Schur convexity and Schur-geometrically concavity of generalized exponent mean, Journal of Mathematical Inequalities, 3, 2, 217-225 (2009) · Zbl 1177.26040
[28] Toader, S.; Toader, G., Complementaries of Greek means with respect to Gini means, International Journal of Applied Mathematics & Statistics, 11, N07, 187-192 (2007) · Zbl 1157.26324
[29] Xia, W.-F.; Chu, Y.-M., Schur-convexity for a class of symmetric functions and its applications, Journal of Inequalities and Applications, 2009 (2009) · Zbl 1187.26006 · doi:10.1155/2009/493759
[30] Shaked, M.; Shanthikumar, J. G.; Tong, Y. L., Parametric Schur convexity and arrangement monotonicity properties of partial sums, Journal of Multivariate Analysis, 53, 2, 293-310 (1995) · Zbl 0827.60011 · doi:10.1006/jmva.1995.1038
[31] Shi, H.-N.; Jiang, Y.-M.; Jiang, W.-D., Schur-convexity and Schur-geometrically concavity of Gini means, Computers & Mathematics with Applications, 57, 2, 266-274 (2009) · Zbl 1165.26311 · doi:10.1016/j.camwa.2008.11.001
[32] Forcina, A.; Giovagnoli, A., Homogeneity indices and Schur-convex functions, Statistica, 42, 4, 529-542 (1982) · Zbl 0506.62035
[33] Yang, Zh.-H., Necessary and sufficient condition for Schur convexity of the two-parameter symmetric homogeneous means, Applied Mathematical Sciences, 5, 61-64, 3183-3190 (2011) · Zbl 1250.26029
[34] Yang, Zh.-H., Necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means, Abstract and Applied Analysis, 2010 (2010) · Zbl 1190.26004 · doi:10.1155/2010/830163
[35] Yang, Zh.-H., Schur harmonic convexity of Gini means, International Mathematical Forum. Journal for Theory and Applications, 6, 13-16, 747-762 (2011) · Zbl 1252.26031
[36] Shi, H. N., Theory of Majorization and Analytic Inequalities (2013), Harbin, China: Harbin Institute of Technology Press, Harbin, China
[37] Shan, Z., Geometric Inequalities (in Chinese) (1980), Shanghai, China: Shanghai Education Press, Shanghai, China
[38] Mitrinović, D. S.; Pečarić, J. E.; Volenec, V., Recent Advances in Geometric Inequalities, 28 (1989), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0679.51004
[39] Zhang, J. Z.; Yang, L., A class of geometric inequalities concerning systems of mass points, Journal of China University of Science and Technology, 11, 2, 1-8 (1981)
[40] Leng, G. S.; Ma, T. Y.; Qian, X. Z., Inequalities for a simplex and an interior point, Geometriae Dedicata, 85, 1-3, 1-10 (2001) · Zbl 0990.51009 · doi:10.1023/A:1010353523857
[41] Yang, S. G.; Wang, W.; Pan, J. J.; Qian, D., The \(n\)-dimensional Pedoe inequality in hyperbolic space with applications, Advances in Mathematics, 41, 3, 347-355 (2012) · Zbl 1274.51019
[42] Xiao, J. Z.; Zhu, X. H., Schur convex functions and inequalities of \(n\)-simplices, Applied Mathematics. A Journal of Chinese Universities A, 16, 4, 428-434 (2001) · Zbl 0992.51009
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