AIMS Mathematics, 2020, 5(4): 2967-2978. doi: 10.3934/math.2020191.

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Refinements of Huygens- and Wilker- type inequalities

1 Department of Mathematics, Zhejiang Gongshang University, Hangzhou, Zhejiang, China
2 Department of Mathematics, Hong Kong Baptist University, Hong Kong
3 Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, China

In this paper we give some refinements and sharpness of the Huygens- and Wilker- type inequalities, and show a proof of the second conjecture by Chen and Chueng in [10].
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Keywords circular functions; hyperbolic functions; refinements and sharpness of the Huygens- and Wilker- type inequalities; proof of the second conjecture by Chen and Chueng

Citation: Ling Zhu, Zhengjie Sun. Refinements of Huygens- and Wilker- type inequalities. AIMS Mathematics, 2020, 5(4): 2967-2978. doi: 10.3934/math.2020191

References

  • 1. F. T. Campan, The Story of Number, Romania, 1977.
  • 2. C. Mortici, The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal. Appl., 14 (2011), 535-541.
  • 3. J. B. Wilker, Problem E 3306, Amer. Math. Monthly, 96 (1989), 55.
  • 4. Z.-H. Yang and Y.-M. Chu, Sharp Wilker-type inequalities with applications, J. Inequal. Appl., 2014 (2014), 166.
  • 5. H.-H. Chu, Z.-H. Yang, Y.-M. Chu, et al. Generalized Wilker-type inequalities with two parameters, J. Inequal. Appl., 2016 (2016), 187.
  • 6. H. Sun, Z.-H. Yang and Y.-M. Chu, Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities, J. Inequal. Appl., 2016 (2016), 322.
  • 7. E. Neuman, On Wilker and Huygens type inequalities, Math. Inequal. Appl., 15 (2012), 271-279.
  • 8. J. S. Sumner, A. A. Jagers, M. Vowe, et al. Inequalities involving trigonometric functions, Amer. Math. Monthly, 98 (1991), 264-267.    
  • 9. W.-D. Jiang, Q.-M. Luo, F. Qi, Refinements and sharpening of some Huygens and Wilker type inequalities, Turkish J. Anal. Number Theory, 2 (2014), 134-139.    
  • 10. Ch.-P. Chen, W.-S. Cheung, Sharpness of Wilker and Huygens Type Inequalities, J. Inequal. Appl., 2012 (2012), 72.
  • 11. J.-L. Li, An identity related to Jordan's inequality, Int. J. Math. Math. Sci., 2006.
  • 12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U. S. National Bureau of Standards, Washington, DC, USA, 1964.
  • 13. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier Academic Press, San Diego, Calif, USA, 3rd edition, 2004.
  • 14. C. D'Aniello, On some inequalities for the Bernoulli numbers, Rendiconti del Circolo Matematico di Palermo. Serie II, 43 (1994), 329-332.    
  • 15. H. Alzer, Sharp bounds for the Bernoulli numbers, Archiv der Mathematik, 74 (2000), 207-211.    
  • 16. Z.-H. Yang and J.-F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math., 364 (2020), 112359.
  • 17. F. Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math., 351 (2019), 1-5.
  • 18. L. Zhu, New bounds for the ratio of two adjacent even-indexed Bernoulli numbers, RACSAM, 114 (2020), 83.

 

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