Abstract and Applied Analysis

A Note on Jordan, Adamović-Mitrinović, and Cusa Inequalities

Zhen-Hang Yang and Yu-Ming Chu

Full-text: Open access

Abstract

We improve the Jordan, Adamović-Mitrinović, and Cusa inequalities. As applications, several new Shafer-Fink type inequalities for inverse sine function and bivariate means inequalities are established, and a new estimate for sine integral is given.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 364076, 12 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412273288

Digital Object Identifier
doi:10.1155/2014/364076

Mathematical Reviews number (MathSciNet)
MR3193504

Zentralblatt MATH identifier
07022231

Citation

Yang, Zhen-Hang; Chu, Yu-Ming. A Note on Jordan, Adamović-Mitrinović, and Cusa Inequalities. Abstr. Appl. Anal. 2014 (2014), Article ID 364076, 12 pages. doi:10.1155/2014/364076. https://projecteuclid.org/euclid.aaa/1412273288


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References

  • D. S. Mitrinović, Analytic Inequalities, vol. 165, Springer, New York, NY, USA, 1970.
  • F. Qi, D.-W. Niu, and B.-N. 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.
  • C.-P. Chen and W.-S. Cheung, “Sharp Cusa and Becker-Stark inequalities,” Journal of Inequalities and Applications, vol. 2011, article 136, 6 pages, 2011.
  • K. S. K. Iyengar, B. S. M. Rao, and T. S. Nanjundiah, “Some trigonometrical inequalities,” Half-Yearly Journal of Mysore University B, vol. 6, pp. 1–12, 1945.
  • C. Mortici, “The natural approach of Wilker-Cusa-Huygens inequalities,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 535–541, 2011.
  • 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.
  • E. Neuman, “Inequalities for the Schwab-Borchardt mean and their applications,” Journal of Mathematical Inequalities, vol. 5, no. 4, pp. 601–609, 2011.
  • J. Sándor and M. Bencze, “On Huygens' trigonometric inequality,” RGMIA Research Report Collection, vol. 8, no. 3, article 14, 2005.
  • 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.
  • Z.-H. Yang, “Refinements of Mitrinović-Cusa inequality,” http://arxiv.org/abs/1206.4911.
  • Z.-H. Yang, “Renements of a two-sided inequality for trigonometric functions,” Journal of Mathematical Inequalities, vol. 7, no. 4, pp. 601–615, 2013.
  • Z.-H. Yang, “New sharp Jordan type inequalities and their applications,” Gulf Journal of Mathematics, vol. 2, no. 1, pp. 1–10, 2014.
  • L. Zhu, “Some new wilker-type inequalities for circular and hyperbolic functions,” Abstract and Applied Analysis, vol. 2009, Article ID 485842, 9 pages, 2009.
  • L. Zhu, “A source of inequalities for circular functions,” Computers & Mathematics with Applications, vol. 58, no. 10, pp. 1998–2004, 2009.
  • C. Huygens, Oeuvres Completes 1888–1940, Sociéte Hollondaise des Science, Haga, Sweden.
  • A. Oppenheim, “Problems for solutions E1277,” The American Mathematical Monthly, vol. 64, no. 7, pp. 504–505, 1957.
  • W. B. Carver, “Extreme parameters in an inequality,” The American Mathematical Monthly, vol. 65, no. 3, pp. 206–209, 1958.
  • Y.-F. Wu, “Remarks on generalizations of Jordan and Kober's inequalities,” Studies in College Mathematics, vol. 11, no. 4, pp. 38–39, 42, 2008 (Chinese).
  • W. D. Jiang, “On Shafer-Fink inequality and Carlson inequality,” The College Mathematics Journal, vol. 23, no. 4, pp. 152–154, 2007 (Chinese).
  • M. Li and D. He, “Better estimations for upper and lower bounds of sin $x/x$ and its application,” Journal of Beijing Union University, vol. 24, no. 2, pp. 47–48, 2010 (Chinese).
  • R. E. Shafer, “An inequality for the inverse tangent E1867,” The American Mathematical Monthly, vol. 74, no. 6, pp. 726–727, 1967.
  • A. M. Fink, “Two inequalities,” Univerzitet u Beogradu. Publikacije Elektrotehnickog Fakulteta. Serija Matematika, vol. 6, pp. 48–49, 1995.
  • L. Zhu, “On Shafer-Fink inequalities,” Mathematical Inequalities & Applications, vol. 8, no. 4, pp. 571–574, 2005.
  • H.-J. Seiffert, “Werte zwischen dem geometrischen und dem arithmetischen Mittel zweier Zahlen,” Elemente der Mathematik, vol. 42, no. 4, pp. 105–107, 1987.
  • H. -J. Seiffert, “Aufgabe $\beta $ 16,” Die Wurzel, vol. 29, pp. 221–222, 1995.
  • E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,” Mathematica Pannonica, vol. 14, no. 2, pp. 253–266, 2003.
  • 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.
  • Y.-M. Chu, Y.-F. Qiu, M.-K. Wang, and G.-D. Wang, “The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert's mean,” Journal of Inequalities and Applications, vol. 2010, Article ID 436457, 7 pages, 2010.
  • Y.-M. Chu, M.-K. Wang, and Y.-F. Qiu, “An optimal double inequality between power-type Heron and Seiffert means,” Journal of Inequalities and Applications, vol. 2010, Article ID 146945, 11 pages, 2010.
  • Y.-M. Chu, M.-K. Wang, and W.-M. Gong, “Two sharp double inequalities for Seiffert mean,” Journal of Inequalities and Applications, vol. 2011, article 44, 7 pages, 2011.
  • 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, 13 pages, 2013.
  • I. Costin and G. Toader, “A nice separation of some Seiffert-type means by power means,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 430692, 6 pages, 2012.
  • A. Witkowski, “Interpolations of Schwab-Borchardt mean,” Mathematical Inequalities & Applications, vol. 16, no. 1, pp. 193–206, 2013.
  • Z.-H. Yang, “Sharp bounds for the second Seiffert mean in terms of power čommentComment on ref. [30?]: Please update the information of these references [30, 31, 33?], if possible.means,” http://arxiv.org/abs/1206.5494.
  • Z.-H. Yang, “The monotonicity results and sharp inequalities for some power-type means of two arguments,” http://arxiv.org/abs/1210.6478v1.
  • Z.-H. Yang, “Estimates for Neuman-Sándor mean by power means and their relative errors,” Journal of Mathematical Inequalities, vol. 7, no. 4, pp. 711–726, 2013.
  • F. Qi, “Extensions and sharpenings of Jordan's and Kober's inequality,” Journal of Mathematics for Technology, vol. 12, no. 4, pp. 98–102, 1996 (Chinese).
  • S. 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.
  • S.-H. Wu, “Sharpness and generalization of Jordan's inequality and its application,” Taiwanese Journal of Mathematics, vol. 12, no. 2, pp. 325–336, 2008. \endinput