Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions
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Yogesh J. Bagul
yjbagul@gmail.com
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Christophe Chesneau
christophe.chesneau@unicaen.fr
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DOI:
https://doi.org/10.4067/S0719-06462019000100021Abstract
The prime goal of this paper is to establish sharp lower and upper bounds for useful functions such as the exponential functions, with a focus on exp(−x2), the trigonometric functions (cosine and sine) and the hyperbolic functions (cosine and sine). The bounds obtained for hyperbolic cosine are very sharp. New proofs, refinements as well as new results are offered. Some graphical and numerical results illustrate the findings.
Keywords
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[13] C. Huygens, Oeuvres completes, Soci Ìet Ìe Hollondaise des Sciences, Haga, 1888-1940.
[14] Y. Lv, G. Wang and Y. Chu, A note on Jordan type inequalities for hyperbolic functions , Appl. Math. Lett., Volume 25, Number 3, pp. 505-508, 2012, doi: 10.1016/j.aml.2011.09.046. [Online]. Available: https://doi.org/10.1016/j.aml.2011.09.046
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[16] E. Neuman and J. Sa Ìndor, On some inequalities involving trigonometric and hyperbolic func- tions with emphasis on the Cusa-Huygens, Wilker and Huygens inequalities, Math. Inequal. Appl., volume 13 Number 4, pp. 715-723, 2010, doi: 10.7153/mia-13-50. [Online]. Available: http://dx.doi.org/10.7153/mia-13-50
[17] F. Qi, D.-W. Niu and B.-N. Guo, Refinements, generalizations and applications of Jor- dan‘s inequality and related problems, Journal of Inequalities and Applications, Volume 2009, Article ID 271923, 52 pages, 2009, doi: 10.1155/2009/271923. [Online]. Available: https://doi.org/10.1155/2009/271923
[18] J. Sa Ìndor, Sharp Cusa-Huygens and related inequalities, Notes on Number Theory and Dis- crete Mathematics, volume 19, Number 1, pp. 50-54, 2013.
[19] J. Sa Ìndor and R. Ol Ìah-Ga Ìl, On Cusa-Huygens type trigonometric and hyperbolic inequalities, Acta Univ. Sapientiae, Mathematica, Volume 4, Number 2, pp. 145-153, 2012.
[20] Z.-H. Yang and Y.-M. Chu, Jordan type inequalities for hyperbolic functions and their ap- plications, Journal of Function Spaces, Volume 2015, Article ID 370979, 4 pages, 2015, doi: 10.1155/2015/370979. [Online]. Available: http://dx.doi.org/10.1155/2015/370979
[21] L. Zhu, A source of inequalities for circular functions, Computers and Mathematics with Applications, Volume 58, Number 10, pp. 1998-2004, 2009, doi: 10.1016/j.camwa.2009.07.076. [Online]. Available: https://doi.org/10.1016/j.camwa.2009.07.076.
[2] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Conformal Invarients, Inequalities and Quasiconformal maps, John Wiley and Sons, New York, 1997.
[3] Y. J. Bagul, On exponential bounds of hyperbolic cosine, Bulletin Of The International Math- ematical Virtual Institute, Volume 8 , Number 2, pp. 365-367, 2018.
[4] Y. J. Bagul, New inequalities involving circular, inverse circular, hyperbolic, inverse hyperbolic and exponential functions, Advances in Inequalities and Applications, Vol- ume 2018, Article ID 5, 8 pages, 2018, doi: 10.28919/aia/3556. [Online]. Available: https://doi.org/10.28919/aia/3556
[5] Y. J. Bagul, Inequalities involving circular, hyperbolic and exponential functions, J. Math. Inequal, Volume 11, Number 3, pp. 695-699, 2017, doi: 10.7153/jmi-2017-11-55. [Online]. Available: http://dx.doi.org/10.7153/jmi-2017-11-55
[6] Y. J. Bagul, On Simple Jordan type inequalities, Turkish J. Ineq., Volume 3, Number 1, pp. 1-6, 2019.
[7] Y. J. Bagul and C. Chesneau, Some sharp circular and hyperbolic bounds of exp(x2) with Applications, preprint. hal-01915086. [Online]. Available: https://hal.archivesouvertes.fr/hal- 01915086
[8] B. A. Bhayo, R. Kl Ìen and J. Sa Ìndor, New trigonometric and hyperbolic inequali- ties, Miskolc Mathematical Notes, Volume 18, Number 1, pp. 125-137, 2017, doi: 10.18514/MMN.2017.1560. [Online]. Available: https://doi.org/10.18514/MMN.2017.1560
[9] J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemann manifolds, J. Differ. Geom., Number 17, 15-53, 1982.
[10] C. Chesneau, Some tight polynomial-exponential lower bounds for an exponential function, Jordan Journal of Mathematics and Statistics (JJMS), Volume 11, Number 3, pp. 273 294, 2018.
[11] C. Chesneau, On two simple and sharp lower bounds for exp(x2), preprint. hal-01593840. [Online]. Available: http://hal.archives-ouvertes.fr/hal-01593840
[12] C. Chesneau, Y. J. Bagul, A note on some new bounds for trigonometric functions using infinite products, 2018, hal-01934571.
[13] C. Huygens, Oeuvres completes, Soci Ìet Ìe Hollondaise des Sciences, Haga, 1888-1940.
[14] Y. Lv, G. Wang and Y. Chu, A note on Jordan type inequalities for hyperbolic functions , Appl. Math. Lett., Volume 25, Number 3, pp. 505-508, 2012, doi: 10.1016/j.aml.2011.09.046. [Online]. Available: https://doi.org/10.1016/j.aml.2011.09.046
[15] B. Malesevic, T. Lutovac and B. Banjac, One method for proving some classes of exponential analytic inequalities, preprint. arXiv:1811.00748v1. [Online]. Available: https://arxiv.org/abs/1811.00748
[16] E. Neuman and J. Sa Ìndor, On some inequalities involving trigonometric and hyperbolic func- tions with emphasis on the Cusa-Huygens, Wilker and Huygens inequalities, Math. Inequal. Appl., volume 13 Number 4, pp. 715-723, 2010, doi: 10.7153/mia-13-50. [Online]. Available: http://dx.doi.org/10.7153/mia-13-50
[17] F. Qi, D.-W. Niu and B.-N. Guo, Refinements, generalizations and applications of Jor- dan‘s inequality and related problems, Journal of Inequalities and Applications, Volume 2009, Article ID 271923, 52 pages, 2009, doi: 10.1155/2009/271923. [Online]. Available: https://doi.org/10.1155/2009/271923
[18] J. Sa Ìndor, Sharp Cusa-Huygens and related inequalities, Notes on Number Theory and Dis- crete Mathematics, volume 19, Number 1, pp. 50-54, 2013.
[19] J. Sa Ìndor and R. Ol Ìah-Ga Ìl, On Cusa-Huygens type trigonometric and hyperbolic inequalities, Acta Univ. Sapientiae, Mathematica, Volume 4, Number 2, pp. 145-153, 2012.
[20] Z.-H. Yang and Y.-M. Chu, Jordan type inequalities for hyperbolic functions and their ap- plications, Journal of Function Spaces, Volume 2015, Article ID 370979, 4 pages, 2015, doi: 10.1155/2015/370979. [Online]. Available: http://dx.doi.org/10.1155/2015/370979
[21] L. Zhu, A source of inequalities for circular functions, Computers and Mathematics with Applications, Volume 58, Number 10, pp. 1998-2004, 2009, doi: 10.1016/j.camwa.2009.07.076. [Online]. Available: https://doi.org/10.1016/j.camwa.2009.07.076.
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Published
2019-04-01
How to Cite
[1]
Y. J. Bagul and C. . Chesneau, “Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions”, CUBO, vol. 21, no. 1, pp. 21–35, Apr. 2019.
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