Journal of Applied Mathematics

Bounds for the Arithmetic Mean in Terms of the Neuman-Sándor and Other Bivariate Means

Fan Zhang, Yu-Ming Chu, and Wei-Mao Qian

Full-text: Open access

Abstract

We present the largest values α1, α2, and α3 and the smallest values β1, β2, and β3 such that the double inequalities α1M(a,b)+(1-α1)H(a,b)<A(a,b)<β1M(a,b)+(1-β1)H(a,b), α2M(a,b)+(1-α2)H-(a,b)<A(a,b)<β2M(a,b)+(1-β2)H-(a,b), and α3M(a,b)+(1-α3)He(a,b)<A(a,b)<β3M(a,b)+(1-β3)He(a,b) hold for all a,b>0 with ab, where M(a,b), A(a,b), He(a,b), H(a,b) and H-(a,b) denote the Neuman-Sándor, arithmetic, Heronian, harmonic, and harmonic root-square means of a and b, respectively.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 582504, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808318

Digital Object Identifier
doi:10.1155/2013/582504

Mathematical Reviews number (MathSciNet)
MR3147875

Zentralblatt MATH identifier
06950758

Citation

Zhang, Fan; Chu, Yu-Ming; Qian, Wei-Mao. Bounds for the Arithmetic Mean in Terms of the Neuman-Sándor and Other Bivariate Means. J. Appl. Math. 2013 (2013), Article ID 582504, 7 pages. doi:10.1155/2013/582504. https://projecteuclid.org/euclid.jam/1394808318


Export citation

References

  • E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,” Mathematica Pannonica, vol. 14, no. 2, pp. 253–266, 2003.
  • 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 a certain bivariate mean,” Journal of Mathematical Inequalities, vol. 6, no. 4, pp. 637–643, 2012.
  • T.-H. Zhao, Y.-M. Chu, and B.-Y. Liu, “Optimal bounds for Neuman-Sándor mean in terms of the convex combinations of harmonic, geometric, quadratic, and contraharmonic means,” Abstract and Applied Analysis, vol. 2012, Article ID 302635, 9 pages, 2012.
  • Y.-M. Chu, B.-Y. Long, W.-M. Gong, and Y.-Q. Song, “Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means,” Journal of Inequalities and Applications, vol. 2013, article 10, 2013.
  • Y.-M. Chu and B.-Y. Long, “Bounds of the Neuman-Sándor mean using power and identric means,” Abstract and Applied Analysis, vol. 2013, Article ID 832591, 6 pages, 2013.
  • T.-H. Zhao, Y.-M. Chu, Y.-L. Jiang, and Y.-M. Li, “Best possible bounds for Neuman-Sándor mean by the identric, quadratic and contraharmonic means,” Abstract and Applied Analysis, vol. 2013, Article ID 348326, 12 pages, 2013.
  • Z.-Y. He, W.-M. Qian, Y.-L. Jiang, Y.-Q. Song, and Y.-M. Chu, “Bounds for the combinations of Neuman-Sándor, arithmetic, and second Seiffert means in terms of contraharmonic mean,” Abstract and Applied Analysis, vol. 2013, Article ID 903982, 5 pages, 2013.
  • W.-M. Qian and Y.-M. Chu, “On certain inequalities for Neuman-Sándor mean,” Abstract and Applied Analysis, vol. 2013, Article ID 790783, 6 pages, 2013.
  • J. Sándor, “On certain inequalities for means III,” Archiv der Mathematik, vol. 76, no. 1, pp. 34–40, 2001.
  • S. Simić and M. Vuorinen, “Landen inequalities for zero-balanced hypergeometric functions,” Abstract and Applied Analysis, vol. 2012, Article ID 932061, 11 pages, 2012. \endinput