Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions

Bagul, Yogesh J.; Chesneau, Christophe; Bagul, Yogesh J.; Chesneau, Christophe

doi:10.4067/S0719-06462019000100021

Services on Demand

Journal

Article

Automatic translation

Indicators

Cited by SciELO
Access statistics

Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.21 no.1 Temuco Apr. 2019

http://dx.doi.org/10.4067/S0719-06462019000100021

Articles

Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions

Yogesh J. Bagul¹

Christophe Chesneau²

^¹K. K. M. College Manwath, Department of Mathematics, Parbhani (M.S.) - 431505, India. yjbagul@gmail.com

^²University of Caen Normandie, LMNO, France christophe.chesneau@unicaen.fr

Abstract

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 and Phrases: Exponential function; trigonometric function; hyperbolic function

Resumen

El objetivo principal de este artículo es establecer cotas inferiores y superiores precisas para funciones útiles tales como las funciones exponenciales, con énfasis especial en exp(−x²), las funciones trigonométricas (coseno y seno) y las funciones hiperbólicas (coseno y seno). Las cotas obtenidas para el coseno hiperbólico son muy precisas. Se presentan, tanto nuevas demostraciones y refinamientos, como resultados nuevos. Algunos resultados numéricos y gráficos ilustran los resultados encontrados.

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

Acknowledgments

We would like to thank the referee for the thorough comments which have helped the presentation of the paper.

References

[1] H. Alzer and M. K. Kwong, On Jordan’s inequality, Period Math Hung, Volume 77, Number 2, pp. 191-200, 2018, doi: 10.1007/s10998-017-0230-z. [Online]. Available: https://doi.org/10.1007/s10998-017-0230z [ Links ]

[2] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Conformal Invarients, Inequalities and Quasiconformal maps, John Wiley and Sons, New York, 1997. [ Links ]

[3] Y. J. Bagul, On exponential bounds of hyperbolic cosine, Bulletin Of The International Mathematical Virtual Institute, Volume 8 , Number 2, pp. 365-367, 2018. [ Links ]

[4] Y. J. Bagul, New inequalities involving circular, inverse circular, hyperbolic, inverse hyperbolic and exponential functions, Advances in Inequalities and Applications, Volume 2018, Article ID 5, 8 pages, 2018, doi: 10.28919/aia/3556. [Online]. Available: https://doi.org/10.28919/aia/3556 [ Links ]

[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 [ Links ]

[6] Y. J. Bagul, On Simple Jordan type inequalities, Turkish J. Ineq., Volume 3, Number 1, pp. 1-6, 2019. [ Links ]

[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.archives-ouvertes.fr/hal-01915086 [ Links ]

[8] B. A. Bhayo, R. Klén and J. Sándor, New trigonometric and hyperbolic inequalities, 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 [ Links ]

[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. [ Links ]

[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. [ Links ]

[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 [ Links ]

[12] C. Chesneau, Y. J. Bagul, A note on some new bounds for trigonometric functions using infinite products, 2018, hal-01934571. [ Links ]

[13] C. Huygens, Oeuvres completes, Société Hollondaise des Sciences, Haga, 1888-1940. [ Links ]

[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 [ Links ]

[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 [ Links ]

[16] E. Neuman andJ. Sándor , On some inequalities involving trigonometric and hyperbolic functions 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 [ Links ]

[17] 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, Volume 2009, Article ID 271923, 52 pages, 2009, doi: 10.1155/2009/271923. [Online]. Available: https://doi.org/10.1155/2009/271923 [ Links ]

[18] J. Sándor , Sharp Cusa-Huygens and related inequalities, Notes on Number Theory and Discrete Mathematics, volume 19, Number 1, pp. 50-54, 2013. [ Links ]

[19] J. Sándor and R. Oláh-Gál, On Cusa-Huygens type trigonometric and hyperbolic inequalities, Acta Univ. Sapientiae, Mathematica, Volume 4, Number 2, pp. 145-153, 2012. [ Links ]

[20] Z.-H. Yang and Y.-M. Chu, Jordan type inequalities for hyperbolic functions and their applications, 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 [ Links ]

[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 [ Links ]

This is an open-access article distributed under the terms of the Creative Commons Attribution License