## Taiwanese Journal of Mathematics

### NOTES ON THE SCHUR-CONVEXITY OF THE EXTENDED MEAN VALUES

#### Abstract

In this article, the Schur-convexities of the weighted arithmetic mean of function and the extended mean values are proved. Moreover, some inequalities involving the arithmetic mean, the harmonic mean, the logarithmic mean, and comparison between the extended mean values and the generalized weighted mean with two parameters and constant weight are obtained.

#### Article information

Source
Taiwanese J. Math., Volume 9, Number 3 (2005), 411-420.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500407849

Digital Object Identifier
doi:10.11650/twjm/1500407849

Mathematical Reviews number (MathSciNet)
MR2162886

Zentralblatt MATH identifier
1086.26008

#### Citation

Qi, Feng; Sándor, József; Dragomir, Sever S.; Sofo, Anthony. NOTES ON THE SCHUR-CONVEXITY OF THE EXTENDED MEAN VALUES. Taiwanese J. Math. 9 (2005), no. 3, 411--420. doi:10.11650/twjm/1500407849. https://projecteuclid.org/euclid.twjm/1500407849

#### References

• [1.] Ch.-P. Chen and F. Qi, A new proof for monotonicity of the generalized weighted mean values, Adv. Stud. Contemp. Math. (Kyungshang) 5 (2003), no. 1, 13-16.
• [2.] B.-N. Guo and F. Qi, Inequalities for generalized weighted mean values of convex function, Math. Inequal. Appl. 4 (2001), no. 2, 195-202. RGMIA Res. Rep. Coll. 2 (1999), no. \!7, Art. \!11,\! 1059-1065.\! Available online at http://rgmia.vu.edu.au/v2n7.html.
• [3.] B.-N. Guo, Sh.-Q. Zhang, and F. Qi, Elementary proofs of monotonicity for extended mean values of some functions with two parameters, Shùxué de Shíjiàn yù Rènsh$\overline{i}\,($Math. Practice Theory$)$ \bf29 (1999), no. 2, 169-174. (Chinese)
• [4.] N. Elezović and J. Pečarić, A note on Schur-convex functions, Rocky Mountain J. Math. 30 (2000), no. 3, 853-856.
• [5.] E. B. Leach and M. C. Sholander, Extended mean values, Amer. Math. Monthly 85 (1978), 84-90.
• [6.] E. Leach and M. Sholander, Extended mean values II, J. Math. Anal. Appl. 92 (1983), 207-223.
• [7.] A. W. Marshall and I. Olkin, Inequalities$:$ Theory of Majorization and its Appplications, Academic Press, New York, 1979.
• [8.] Z. Páles, Inequalities for differences of powers, J. Math. Anal. Appl. 131 (1988), 271-281.
• [9.] J. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering 187, Academic Press, 1992.
• [10.] J. Pečarić, F. Qi, V. Šimić and S.-L. Xu, Refinements and extensions of an inequality, I\!I\!I, J. Math. Anal. Appl. 227 (1998), no. 2, 439-448.
• [11.] F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl. Math. \bf1 (2000), no. 1, Art. 4. Available online at http://jipam.vu.edu.au/article.php?sid=97. RGMIA Res. Rep. Coll. \bf2 (1999), no. 5, Art. 4, 633-642. Available online at http://rgmia.vu.edu.au/v2n5.html.
• [12.] F. Qi, Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1978, 2723-2732.
• [13.] F. Qi, Logarithmic convexity of extended mean values, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1787-1796. Logarithmic convexity of the extended mean values, RGMIA Res. Rep. Coll. \bf2 (1999), no. 5, Art. 5, 643-652. Available online at http://rgmia.vu.edu.au/v2n5.html.
• [14.] F. Qi, On a two-parameter family of nonhomogeneous mean values, Tamkang J. Math. 29 (1998), no. 2, 155-163.
• [15.] F. Qi, Schur-convexity of the extended mean values, Rocky Mountain J. Math. 35 (2005), in press. RGMIA Res. Rep. Coll. \bf4 (2001), no. 4, Art. 4, 529-533. Available online at http://rgmia.vu.edu.au/v4n4.html.
• [16.] F. Qi, The extended mean values: definition, properties, monotonicities, comparison, convexities, generalizations, and applications, Cubo Mat. Educ. 5 (2003), no. 3, 63-90. RGMIA Res. Rep. Coll. \bf5 (2002), no. 1, Art. 5, 57-80. Available online at http://rgmia.vu.edu.au/v5n1.html.
• [17.] F. Qi and J.-X. Cheng, Some new Steffensen pairs, Anal. Math. 29 (2003), 219-226. New Steffensen pairs, RGMIA Res. Rep. Coll. 3 (2000), no. 3, Art. 11, 431-436. Available online at http://rgmia.vu.edu.au/v3n3.html.
• [18.] F. Qi and B.-N. Guo, On Steffensen pairs, J. Math. Anal. Appl. 271 (2002), no. 2, 534-541. RGMIA Res. Rep. Coll. 3 (2000), no. 3, Art. 10, 425-430. Available online at http://rgmia.vu.edu.au/v3n3.html.
• [19.] F. Qi and Q.-M. Luo, A simple proof of monotonicity for extended mean values, J. Math. Anal. Appl. 224 (1998), no. 2, 356-359.
• [20.] F. Qi, J.-Q. Mei, D.-F. Xia, and S.-L. Xu, New proofs of weighted power mean inequalities and monotonicity for generalized weighted mean values, Math. Inequal. Appl. \bf3 (2000), no. 3, 377-383.
• [21.] F. Qi, J.-Q. Mei and S.-L. Xu, Other proofs of monotonicity for generalized weighted mean values, RGMIA Res. Rep. Coll. 2 (1999), no. 4, Art. 6, 469-472. Available online at http://rgmia.vu.edu.au/v2n4.html.
• [22.] F. Qi, J. Sándor, S. S. Dragomir, and A. Sofo, Notes on the Schur-convexity of the extended mean values, RGMIA Res. Rep. Coll. 5 (2002), no. 1, Art. 3, 19-27. Available online at http://rgmia.vu.edu.au/v5n1.html.
• [23.] F. Qi and N. Towghi, Inequalities for the ratios of the mean values of functions, Nonlinear Funct. Anal. Appl. 9 (2004), no. 1, 15-23.
• [24.] F. Qi and S.-L. Xu, The function $(b^x-a^x)/x$: Inequalities and properties, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3355-3359.
• [25.] F. Qi, S.-L. Xu, and L. Debnath, A new proof of monotonicity for extended mean values, Internat. J. Math. Math. Sci. 22 (1999), no. 2, 415-420.
• [26.] F. Qi and Sh.-Q. Zhang, Note on monotonicity of generalized weighted mean values, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1989, 3259-3260.
• [27.] J. Sándor, On means generated by derivatives of functions, Internat. J. Math. Ed. Sci. Tech. 28 (1997), 146-148.
• [28.] J. Sándor and Gh. Toader, Some general means, Czechoslovak Math. J. 49 (1999), no. 124, 53-62.
• [28.] K. B. Stolarsky, Generalizations of the logarithmic mean, Mag. Math. 48 (1975), 87-92.
• [29.] N. Towghi and F. Qi, An inequality for the ratios of the arithmetic means of functions with a positive parameter, RGMIA Res. Rep. Coll. 4 (2001), no. 2, Art. 15, 305-309. Available online at http://rgmia.vu.edu.au/v4n2.html.
• [30.] D.-F. Xia, S.-L. Xu and F. Qi, A proof of the arithmetic mean-geometric mean-harmonic mean inequalities, RGMIA Res. Rep. Coll. 2 (1999), no. 1, Art. 10, 99-102. Available online at http://rgmia.vu.edu.au/v2n1.html.