THE STOLARSKY TYPE FUNCTIONS AND THEIR MONOTONICITIES

Year 2009, Volume: 38 Issue: 2, 119 - 128, 01.02.2009

V. Lokesha Z.-g. Wang Z.-h. Zhang S. Padmanabhan

Abstract

In this paper, we give the definition of a Stolarsky type function, and obtain its monotonicity. By using these results, we establish a series of means and their monotonicities in n variables.

Keywords

Two-parameter, Monotonicity, Mean, Vandermonde determinant, 2000 AMS Classification: 26 D 15

THE STOLARSKY TYPE FUNCTIONS AND THEIR MONOTONICITIES

Abstract

References

• Hardy, G. H., Littlewood, J. E. and Polya, G. Inequalities (2nd.ed., Cambridge University Press, Cambrige, 1959).
• Kuang, J. -C. Applied Inequalities (Hunan Education Press, 2nd. Ed., 1993), 39–49. (in Chinese).
• Leach, E. B. and Sholander, M. C. Extended mean values, Amer. Math. Monthly 85, 84–90, 1978.
• Leach, E. B. and Sholander, M. C. Extended mean values, J. Math. Anal. Appl. 92, 207–223, 1983.
• Lokesha, V. and Zhang, Z. -H. The weighted Heron dual mean in n variables, Adv. Stud. Contemp. Math. (Kyungshang) 13 (2), 165–170, 2006.
• Pe˘cari´c, J. and ˘Simi´c, V. Stolarsky-Tobey mean in n variables, Math. Inequal. Appl. 2 (3), 328–341, 1999. [7] Qi, F. Generalized weighted mean values with two parameters, Proc. Royal Soc. London, Series, Math. Physical Engineering Sci. 454, 2723–2732, 1998.
• Qi, F. On a two-parameter family of nonhomogeneous mean values, Tamkang J. Math. 29, 155–163, 1998. [9] Stolarsky, K. B. Generalizations of the logarithmic mean, Math. Mag. 48, 87–92, 1975.
• Xiao, Z. -G. and Zhang, Z. -H. The Henleman mean of n positive numbers, J. Yueyang Normal Univ. 14 (2), 1–5, 2001 (in Chinese).
• Xiao, Z. -G. and Zhang, Z. -H. The Stolarsky mean of n positive numbers, J. Yueyang Normal Uinv. 14 (4), 5–8, 2001 (in Chinese).
• Xiao, Z. -G., Zhang, Z. -H., Lokesha, V. and Nagaraja, K. M. A class of new three-parameter generalized weighted means, Int. J. Appl. Math. Stat. 11 (7), 193–202, 2007.
• Xiao, Z. -G, Zhang, Z. -H., Lokesha, V. and Tang, R. The two-parameter mean of n variables, Int. Rev. Pure Appl. Math. 1 (1), 93–111, 2005.
• Xiao, Z. -G., Zhang, Z. -H. and Qi, F. A type of mean values of several positive numbers with two parameters, Nonlinear Funct. Anal. Appl. 12, 687–702, 2007.
• Zhang, Z. -H. Three classes of new means in n+1 variables and their applications, J. Hunan Ed. Inst. 15 (5), 130–136, 1997 (in Chinese).
• Zhang, Z. -H., Lokesha, V. and Xiao, Z. -G. The weighted Heron mean in n variables, J. Anal. Comput. 1 (1), 57–68, 2005.
• Zhang, Z. -H., Xiao, Z. -G. and Srivastava, H. M. A general family of weighted elementary symmetric means, Appl. Math. Lett. (2008), doi:10.1016/j.aml.2007.12.030