Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2015 (2015), Article ID 517647, 5 pages
http://dx.doi.org/10.1155/2015/517647
Research Article

Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

1School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China
2Department of Mathematics, Huzhou University, Huzhou 313000, China

Received 3 November 2014; Accepted 27 April 2015

Academic Editor: David R. Larson

Copyright © 2015 Yu-Ming Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, D. Reidel Publishing Company, Dordrecht, The Netherlands, 1988. View at MathSciNet
  2. C. O. Imoru, “The power mean and the logarithmic mean,” International Journal of Mathematics and Mathematical Sciences, vol. 5, no. 2, pp. 337–343, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. E. Pečarić, “Generalization of the power means and their inequalities,” Journal of Mathematical Analysis and Applications, vol. 161, no. 2, pp. 395–404, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. D. Lukkassen, “Means of power type and their inequalities,” Mathematische Nachrichten, vol. 205, pp. 131–147, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. Z. Liu, “Remark on inequalities between Hölder and Lehmer means,” Journal of Mathematical Analysis and Applications, vol. 247, no. 1, pp. 309–313, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  6. H. Alzer, “A power mean inequality for the gamma function,” Monatshefte für Mathematik, vol. 131, no. 3, pp. 179–188, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  7. C. Mortici, “Arithmetic mean of values and value at mean of arguments for convex functions,” The ANZIAM Journal, vol. 50, no. 1, pp. 137–141, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. C. Mortici, “A power series approach to some inequalities,” The American Mathematical Monthly, vol. 119, no. 2, pp. 147–151, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. L.-M. Zhou, S.-L. Qiu, and F. Wang, “Inequalities for the generalized elliptic integrals with respect to Hölder means,” Journal of Mathematical Analysis and Applications, vol. 386, no. 2, pp. 641–646, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. B. A. Bhayo, “On the power mean inequality of the hyperbolic metric of unit ball,” The Journal of Prime Research in Mathematics, vol. 8, pp. 45–50, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  11. T. P. Lin, “The power mean and the logarithmic mean,” The American Mathematical Monthly, vol. 81, pp. 879–883, 1974. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,” Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, vol. 678–715, pp. 15–18, 1980. View at Google Scholar
  13. A. A. Jagers, “Solutions of problem 887,” Nieuw Archief voor Wiskunde, vol. 12, no. 4, pp. 230–231, 1994. View at Google Scholar
  14. H. J. Seiffert, “Aufgabe β16,” Die Wurzel, vol. 29, pp. 221–222, 1995. View at Google Scholar
  15. H. Alzer, “Ungleichungen für Mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422–426, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. H. Alzer and S.-L. Qiu, “Inequalities for means in two variables,” Archiv der Mathematik, vol. 80, no. 2, pp. 201–215, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. I. Costin and Gh. Toader, “A separation of some Seiffert-type means by power means,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 41, no. 2, pp. 125–129, 2012. View at Google Scholar · View at MathSciNet
  18. P. A. Hastö, “Optimal inequalities between Seiffert's mean and power means,” Mathematical Inequalities & Applications, vol. 7, no. 1, pp. 47–53, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  19. I. Costin and Gh. Toader, “Optimal evaluations of some Seiffert-type means by power means,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4745–4754, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. Y.-M. Li, M.-K. Wang, and Y.-M. Chu, “Sharp power mean bounds for Seiffert mean,” Applied Mathematics, vol. 29, no. 1, pp. 101–107, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. A. Čizmesija, “A new sharp double inequality for generalized Heronian, harmonic and power means,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 664–671, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. M.-K. Wang, Y.-M. Chu, Y.-F. Qiu, and S.-L. Qiu, “An optimal power mean inequality for the complete elliptic integrals,” Applied Mathematics Letters, vol. 24, no. 6, pp. 887–890, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. Y.-M. Chu, S.-L. Qiu, and M.-K. Wang, “Sharp inequalities involving the power mean and complete elliptic integral of the first kind,” The Rocky Mountain Journal of Mathematics, vol. 43, no. 5, pp. 1489–1496, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. E. Neuman, “A one-parameter family of bivariate means,” Journal of Mathematical Inequalities, vol. 7, no. 3, pp. 399–412, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. Y.-M. Chu, M.-K. Wang, and Z.-K. Wang, “A best-possible double inequality between Seiffert and harmonic means,” Journal of Inequalities and Applications, vol. 2011, article 94, 7 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Y.-M. Chu, M.-K. Wang, and Z.-K. Wang, “Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means,” Mathematical Inequalities & Applications, vol. 15, no. 2, pp. 415–422, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  27. Y.-M. Chu and S.-W. Hou, “Sharp bounds for Seiffert mean in terms of contraharmonic mean,” Abstract and Applied Analysis, vol. 2012, Article ID 425175, 6 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus