Abstract and Applied Analysis

Optimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means

Tie-Hong Zhao, Yu-Ming Chu, and Bao-Yu Liu

Full-text: Open access

Abstract

We present the best possible lower and upper bounds for the Neuman-Sándor mean in terms of the convex combinations of either the harmonic and quadratic means or the geometric and quadratic means or the harmonic and contraharmonic means.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 302635, 9 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475996

Digital Object Identifier
doi:10.1155/2012/302635

Mathematical Reviews number (MathSciNet)
MR3004894

Zentralblatt MATH identifier
1256.26018

Citation

Zhao, Tie-Hong; Chu, Yu-Ming; Liu, Bao-Yu. Optimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means. Abstr. Appl. Anal. 2012 (2012), Article ID 302635, 9 pages. doi:10.1155/2012/302635. https://projecteuclid.org/euclid.aaa/1364475996


Export citation

References

  • E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,” Mathematica Pannonica, vol. 14, no. 2, pp. 253–266, 2003.
  • Y.-M. Chu and M.-K. Wang, “Inequalities between arithmetic-geometric, Gini, and Toader means,” Abstract and Applied Analysis, vol. 2012, Article ID 830585, 11 pages, 2012.
  • E. Neuman, “On one-parameter family of bivariate means,” Aequationes Mathematicae, vol. 83, no. 1-2, pp. 191–197, 2012.
  • M.-K. Wang, Z.-K. Wang, and Y.-M. Chu, “An optimal double inequality between geometric and identric means,” Applied Mathematics Letters, vol. 25, no. 3, pp. 471–475, 2012.
  • W.-D. Jiang, “Some sharp inequalities involving reciprocals of the Seiffert and other means,” Journal of Mathematical Inequalities, vol. 6, no. 4, pp. 593–599, 2012.
  • E. Neuman, “Inequalities for the Schwab-Borchardt mean and their applications,” Journal of Mathematical Inequalities, vol. 5, no. 4, pp. 601–609, 2011.
  • E. Neuman and J. Sándor, “Companion inequalities for certain bivariate means,” Applicable Analysis and Discrete Mathematics, vol. 3, no. 1, pp. 46–51, 2009.
  • L. Zhu, “Some new inequalities for means in two variables,” Mathematical Inequalities & Applications, vol. 11, no. 3, pp. 443–448, 2008.
  • L. Zhu, “New inequalities for means in two variables,” Mathematical Inequalities & Applications, vol. 11, no. 2, pp. 229–235, 2008.
  • E. Neuman and J. Sándor, “Inequalities for the ratios of certain bivariate means,” Journal of Mathematical Inequalities, vol. 2, no. 3, pp. 383–396, 2008.
  • H. Alzer and S.-L. Qiu, “Inequalities for means in two variables,” Archiv der Mathematik, vol. 80, no. 2, pp. 201–215, 2003.
  • J. Sándor, “On certain inequalities for means III,” Archiv der Mathematik, vol. 76, no. 1, pp. 34–40, 2001.
  • J. Sándor, “On certain inequalities for means II,” Journal of Mathematical Analysis and Applications, vol. 199, no. 2, pp. 629–635, 1996.
  • J. Sándor, “On certain inequalities for means,” Journal of Mathematical Analysis and Applications, vol. 189, no. 2, pp. 602–606, 1995.
  • M. K. Vamanamurthy and M. Vuorinen, “Inequalities for means,” Journal of Mathematical Analysis and Applications, vol. 183, no. 1, pp. 155–166, 1994.
  • J. Sándor, “A note on some inequalities for means,” Archiv der Mathematik, vol. 56, no. 5, pp. 471–473, 1991.
  • P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, D. Reidel, Dordrecht, The Netherlands, 1988.
  • E. Neuman and J. Sándor, “On the Schwab-Borchardt mean. II,” Mathematica Pannonica, vol. 17, no. 1, pp. 49–59, 2006.
  • Y.-M. Li, B.-Y. Long, and Y.-M. Chu, “Sharp bounds for the Neuman-Sándor mean in terms of generalized logarithmic mean,” Journal of Mathematical Inequalities, vol. 6, no. 4, pp. 567–577, 2012.
  • E. Neuman, “A note on certain bivariate mean,” Journal of Mathematical Inequalities, vol. 6, no. 4, pp. 637–643, 2012.
  • S. Simić and M. Vuorinen, “Landen inequalities for zero-balanced hypergeometric functions,” Abstract and Applied Analysis, vol. 2012, Article ID 932061, 11 pages, 2012.