About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 592085, 9 pages
http://dx.doi.org/10.1155/2014/592085
Research Article

A Sharp Double Inequality for Trigonometric Functions and Its Applications

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

Received 26 April 2014; Accepted 20 June 2014; Published 10 July 2014

Academic Editor: Josip E. Pečarić

Copyright © 2014 Zhen-Hang Yang 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. D. S. Mitrinović, Analytic Inequalities, Springer, New York, NY, USA, 1970.
  2. C. Huygens, Oeuvres Completes 1888–1940, Sociéte Hollondaise des Science, Haga, Sweden, 1940.
  3. C. Chen and W. Cheung, “Sharp Cusa and Becker-Stark inequalities,” Journal of Inequalities and Applications, vol. 2011, article 136, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. C. Mortici, “The natural approach of Wilker-Cusa-Huygens inequalities,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 535–541, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. C. Mortici, “A subtly analysis of Wilker inequality,” Applied Mathematics and Computation, vol. 231, pp. 516–520, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  6. E. Neuman and J. Sándor, “On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities,” Mathematical Inequalities & Applications, vol. 13, no. 4, pp. 715–723, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. J. Sándor and M. Bencze, “On Huygen's trigonometric inequality,” RGMIA Research Report Collection, vol. 8, no. 3, article 14, 2005.
  8. L. Zhu, “A source of inequalities for circular functions,” Computers & Mathematics with Applications, vol. 58, no. 10, pp. 1998–2004, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. F. Qi, L. H. Cui, and S. L. Xu, “Some inequalities constructed by Tchebysheff's integral inequality,” Mathematical Inequalities & Applications, vol. 2, no. 4, pp. 517–528, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. Y. Lv, G. Wang, and Y. Chu, “A note on Jordan type inequalities for hyperbolic functions,” Applied Mathematics Letters, vol. 25, no. 3, pp. 505–508, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. E. Neuman, “Refinements and generalizations of certain inequalities involving trigonometric and hyperbolic functions,” Advances in Inequalities and Applications, vol. 1, no. 1, pp. 1–11, 2012.
  12. E. Neuman, “Inequalities for the Schwab-Borchardt mean and their applications,” Journal of Mathematical Inequalities, vol. 5, no. 4, pp. 601–609, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. R. Klén, M. Visuri, and M. Vuorinen, “On Jordan type inequalities for hyperbolic functions,” Journal of Inequalities and Applications, vol. 2010, Article ID 362548, 14 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Z.-H. Yang, “New sharp Jordan type inequalities and their applications,” Gulf Journal of Mathematics, vol. 2, no. 1, pp. 1–10, 2014.
  15. Z.-H. Yang, “Three families of two-parameter means constructed by trigonometric functions,” Journal of Inequalities and Applications, vol. 2013, article 541, 27 pages, 2013. View at Publisher · View at Google Scholar
  16. Z. Yang, “Refinements of a two-sided inequality for trigonometric functions,” Journal of Mathematical Inequalities, vol. 7, no. 4, pp. 601–615, 2013. View at MathSciNet
  17. Zh.-H. Yang, “Sharp bounds for Seiffert mean in terms of weighted power means of arithmetic mean and geometric mean,” Mathematical Inequalities & Applications, vol. 17, no. 2, pp. 499–511, 2014.
  18. K. S. K. Iyengar, B. S. Madhava Rao, and T. S. Nanjundiah, “Some trigonometrical inequalities,” The Half-Yearly Journal of the Mysore University B, vol. 6, pp. 1–12, 1945. View at MathSciNet
  19. S.-H. Wu and L. Debnath, “A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality,” Applied Mathematics Letters, vol. 19, no. 12, pp. 1378–1384, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. S. Wu, “Sharpness and generalization of Jordan's inequality and its application,” Taiwanese Journal of Mathematics, vol. 12, no. 2, pp. 325–336, 2008. View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. S.-H. Wu and Á. Baricz, “Generalizations of Mitrinović, Adamović and Lazarević 's inequalities and their applications,” Publicationes Mathematicae Debrecen, vol. 75, no. 3-4, pp. 447–458, 2009. View at MathSciNet
  22. F. Qi, D. Niu, and B. Guo, “Refinements, generalizations, and applications of Jordan's inequality and related problems,” Journal of Inequalities and Applications, vol. 2009, Article ID 271923, 52 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalties, D. Reidel Publishing, Dordrecht, The Netherlands, 1988.
  24. E. Neuman, “On Wilker and Huygens type inequalities,” Mathematical Inequalities & Applications, vol. 15, no. 2, pp. 271–279, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,” Mathematica Pannonica, vol. 14, no. 2, pp. 253–266, 2003. View at MathSciNet
  26. E. Neuman and J. Sándor, “On the schwab-borchardt mean II,” Mathematica Pannonica, vol. 17, no. 1, pp. 49–59, 2006. View at MathSciNet
  27. E. Neuman, “On some means derived from the Schwab-Borchardt mean,” Journal of Mathematical Inequalities, vol. 8, no. 1, pp. 171–181, 2014.
  28. H. Seiffert, “Werte zwischen dem geometrischen und dem arithmetischen Mittel zweier Zahlen,” Elemente der Mathematik Revue de Math ematiques El ementaires Rivista de Matematica Elementare, vol. 42, no. 4, pp. 105–107, 1987. View at MathSciNet
  29. H.-J. Seiffert, “Aufgabe β 16,” Die Wurzel, vol. 29, pp. 221–222, 1995.
  30. R. E. Shafer, L. S. Grinstein, and D. C. B. Marsh, “Problems and solutions: solutions of elementary problems: E1867,” The American Mathematical Monthly, vol. 74, no. 6, pp. 726–727, 1967. View at MathSciNet
  31. A. M. Fink, “Two inequalities,” Univerzitet u Beogradu. Publikacije Elektrotehni v ckog Fakulteta: Serija Matematika, vol. 6, pp. 48–49, 1995.
  32. L. Zhu, “On Shafer-Fink inequalities,” Mathematical Inequalities and Applications, vol. 8, no. 4, pp. 571–574, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus