Abstract and Applied Analysis

Schur $m$-Power Convexity of a Class of Multiplicatively Convex Functions and Applications

Abstract

We investigate the conditions under which the symmetric functions ${F}_{n,k}(\mathbf{x},r)={\prod }_{1\le {i}_{1}<{i}_{2}<\cdots <{i}_{k}\le n}\mathrm{ }\mathrm{‍}f(({\sum }_{j=1}^{k}\mathrm{‍}{x}_{{i}_{j}}^{r}{)}^{1/r}), k=1,2,\dots ,\mathrm{n,}$ 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.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 258108, 12 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412277005

Digital Object Identifier
doi:10.1155/2014/258108

Mathematical Reviews number (MathSciNet)
MR3214410

Zentralblatt MATH identifier
07022017

Citation

Wang, Wen; Yang, Shiguo. Schur $m$ -Power Convexity of a Class of Multiplicatively Convex Functions and Applications. Abstr. Appl. Anal. 2014 (2014), Article ID 258108, 12 pages. doi:10.1155/2014/258108. https://projecteuclid.org/euclid.aaa/1412277005

References

• P. S. Bullen, D. S. Mitrinovic, and M. Vasic, Handbook of Means and Theirs Inequality, Kluwer Academic, Dordrecht, The Netherlands, 2003.
• T. Hara, M. Uchiyama, and S.-E. Takahasi, “A refinement of various mean inequalities,” Journal of Inequalities and Applications, vol. 2, no. 4, pp. 387–395, 1998.
• K. Z. Guan, “A class of symmetric functions for multiplicatively convex function,” Mathematical Inequalities & Applications, vol. 10, no. 4, pp. 745–753, 2007.
• I. Rovenţa, “Schur convexity of a class of symmetric functions,” Annals of the University of Craiova. Mathematics and Computer Science Series, vol. 37, no. 1, pp. 12–18, 2010.
• J. X. Meng, Y. M. Chu, and X. M. Tang, “The Schur-harmonic-convexity of dual form of the Hamy symmetric function,” Matematichki Vesnik, vol. 62, no. 1, pp. 37–46, 2010.
• H. N. Shi and J. Zhang, “Schur-convexity, Schur-geometric and Harmonic convexity of dual form of a class symmetric functions,” Journal of Inequalities and Applications, vol. 2013, p. 295, 2013.
• Zh.-H. Yang, “Schur power convexity of Stolarsky means,” Publicationes Mathematicae Debrecen, vol. 80, no. 1-2, pp. 43–66, 2012.
• Zh.-H. Yang, “Schur power convexity of Gini means,” Bulletin of the Korean Mathematical Society, vol. 50, no. 2, pp. 485–498, 2013.
• Zh.-H. Yang, “Schur power convexity of the Daróczy means,” Mathematical Inequalities & Applications, vol. 16, no. 3, pp. 751–762, 2013.
• W. Wang and S. G. Yang, “Schur m-power convexityčommentComment on ref. [10?]: Please update the information of this reference, if possible. of generalized Hamy symmetric function,” Journal of Mathematical Inequalities. In press.
• W. Wang and S. G. Yang, “On the Schur m-powerčommentComment on ref. [11?]: Please update the information of this reference, if possible. convexity for a class of symmetric functions,” Journal of Systems Science and Mathematical Sciences. In press.
• B. Y. Wang, Foundations of Majorization Inequalities (in Chinese), Beijing Normal University Press, Beijing, China, 1990.
• A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer, New York, NY, USA, 2nd edition, 2011.
• X. M. Zhang, Geometrically-Convex Functions (in Chinese), Anhui University Press, Hefei, China, 2004.
• Y. M. Chu and T. C. Sun, “The Schur harmonic convexity for a class of symmetric functions,” Acta Mathematica Scientia B, vol. 30, no. 5, pp. 1501–1506, 2010.
• Y. Wu and F. Qi, “Schur-harmonic convexity for differences of some means,” Analysis, vol. 32, no. 4, pp. 263–270, 2012.
• C. P. Niculescu, “Convexity according to the geometric mean,” Mathematical Inequalities & Applications, vol. 3, no. 2, pp. 155–167, 2000.
• B. Y. Long and Y. M. Chu, “The Schur convexity and inequalities for a class of symmetric functions,” Acta Mathematica Scientia A, vol. 32, no. 1, pp. 80–89, 2012 (Chinese).
• S. H. Wu, “Generalization and sharpness of the power means inequality and their applications,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 637–652, 2005.
• W. F. Xia and Y. M. Chu, “Schur convexity with respect to a class of symmetric functions and their applications,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 3, pp. 84–96, 2011.
• H.-T. Ku, M.-C. Ku, and X.-M. Zhang, “Inequalities for symmetric means, symmetric harmonic means, and their applications,” Bulletin of the Australian Mathematical Society, vol. 56, no. 3, pp. 409–420, 1997.
• K. Z. Guan, “The Hamy symmetric function and its generalization,” Mathematical Inequalities & Applications, vol. 9, no. 4, pp. 797–805, 2006.
• G. H. Hardy, J. E. Littlewood, and G. Pólya, “Some simple inequalities satisfied by convex functions,” Messenger of Mathematics, vol. 58, pp. 145–152, 1929.
• Y. M. Chu, W. F. Xia, and T. H. Zhao, “Schur convexity for a class of symmetric functions,” Science China. Mathematics, vol. 53, no. 2, pp. 465–474, 2010.
• F. Qi, J. Sándor, S. S. Dragomir, and A. Sofo, “Notes on the Schur-convexity of the extended mean values,” Taiwanese Journal of Mathematics, vol. 9, no. 3, pp. 411–420, 2005.
• G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Generalized convexity and inequalities,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1294–1308, 2007.
• D.-M. Li and H.-N. Shi, “Schur convexity and Schur-geometrically concavity of generalized exponent mean,” Journal of Mathematical Inequalities, vol. 3, no. 2, pp. 217–225, 2009.
• S. Toader and G. Toader, “Complementaries of Greek means with respect to Gini means,” International Journal of Applied Mathematics & Statistics, vol. 11, no. N07, pp. 187–192, 2007.
• W.-F. Xia and Y.-M. Chu, “Schur-convexity for a class of symmetric functions and its applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 493759, 15 pages, 2009.
• M. Shaked, J. G. Shanthikumar, and Y. L. Tong, “Parametric Schur convexity and arrangement monotonicity properties of partial sums,” Journal of Multivariate Analysis, vol. 53, no. 2, pp. 293–310, 1995.
• H.-N. Shi, Y.-M. Jiang, and W.-D. Jiang, “Schur-convexity and Schur-geometrically concavity of Gini means,” Computers & Mathematics with Applications, vol. 57, no. 2, pp. 266–274, 2009.
• A. Forcina and A. Giovagnoli, “Homogeneity indices and Schur-convex functions,” Statistica, vol. 42, no. 4, pp. 529–542, 1982.
• Zh.-H. Yang, “Necessary and sufficient condition for Schur convexity of the two-parameter symmetric homogeneous means,” Applied Mathematical Sciences, vol. 5, no. 61–64, pp. 3183–3190, 2011.
• Zh.-H. Yang, “Necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means,” Abstract and Applied Analysis, vol. 2010, Article ID 830163, 16 pages, 2010.
• Zh.-H. Yang, “Schur harmonic convexity of Gini means,” International Mathematical Forum. Journal for Theory and Applications, vol. 6, no. 13–16, pp. 747–762, 2011.
• H. N. Shi, Theory of Majorization and Analytic Inequalities, Harbin Institute of Technology Press, Harbin, China, 2013.
• Z. Shan, Geometric Inequalities (in Chinese), Shanghai Education Press, Shanghai, China, 1980.
• D. S. Mitrinović, J. E. Pečarić, and V. Volenec, Recent Advances in Geometric Inequalities, vol. 28, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
• J. Z. Zhang and L. Yang, “A class of geometric inequalities concerning systems of mass points,” Journal of China University of Science and Technology, vol. 11, no. 2, pp. 1–8, 1981.
• G. S. Leng, T. Y. Ma, and X. Z. Qian, “Inequalities for a simplex and an interior point,” Geometriae Dedicata, vol. 85, no. 1–3, pp. 1–10, 2001.
• S. G. Yang, W. Wang, J. J. Pan, and D. Qian, “The $n$-dimensional Pedoe inequality in hyperbolic space with applications,” Advances in Mathematics, vol. 41, no. 3, pp. 347–355, 2012.
• J. Z. Xiao and X. H. Zhu, “Schur convex functions and inequalities of $n$-simplices,” Applied Mathematics. A Journal of Chinese Universities A, vol. 16, no. 4, pp. 428–434, 2001. \endinput