Abstract
In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential inequalities.
Similar content being viewed by others
References
Agarwal, R.P., Kim, Y.H., Sen, S.K.: A new refined Jordan’s inequality and its application. Math. Inequal. Appl. 12(2), 255–264 (2009)
Alirezaei, G., Mathar, R.: Scrutinizing the average error probability for nakagami fading channels. In: The IEEE International Symposium on Information Theory (ISIT’14), Honolulu, Hawai, pp. 2884–2888 (2014)
Chen, X.-D., Shi, J., Wang, Y., Xiang, P.: A new method for sharpening the bounds of several special functions. Results Math. 72(1–2), 695–702 (2017)
Cloud, M.J., Drachman, B.C., Lebedev, L.P.: Inequalities with Applications to Engineering. Springer, Berlin (2014)
Debnath, L., Mortici, C., Zhu, L.: Refinements of Jordan–Stečkin and Becker–Stark inequalities. Results Math. 67(1–2), 207–215 (2015)
Gradshteyn, I., Ryzhik, I.: Table of Integrals Series and Products, 8th edn. Academic Press, Cambridge (2015)
Lutovac, T., Malešević, B., Mortici, C.: The natural algorithmic approach of mixed trigonometric-polynomial problems. J. Inequal. Appl. 2017(116), 1–16 (2017)
Makragić, M.: A method for proving some inequalities on mixed hyperbolic-trigonometric polynomial functions. J. Math. Inequal. 11(3), 817–829 (2017)
Malešević, B., Rašajski, M., Lutovac, T.: Refined estimates and generalizations of inequalities related to the arctangent function and Shafer’s inequality. arXiv:1711.03786
Malešević, B., Makragić, M.: A method for proving some inequalities on mixed trigonometric polynomial functions. J. Math. Inequal. 10(3), 849–876 (2016)
Malešević, B., Rašajski, M., Lutovac, T.: Refinements and generalizations of some inequalities of Shafer–Fink’s type for the inverse sine function. J. Inequal. Appl. 2017(275), 1–9 (2017)
Malešević, B., Lutovac, T., Rašajski, M., Mortici, C.: Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities. Adv. Differ. Equ. 2018(90), 1–15 (2018)
Malešević, B., Lutovac, T., Banjac, B.: A proof of an open problem of Yusuke Nishizawa for a power-exponential function. J. Math. Inequal. 12(2), 473–485 (2018)
Mitrinović, D.S.: Analytic Inequalities. Springer, Berlin (1970)
Mortici, C.: The natural approach of Wilker–Cusa–Huygens inequalities. Math. Inequal. Appl. 14(3), 535–541 (2011)
Nenezić, M., Malešević, B., Mortici, C.: New approximations of some expressions involving trigonometric functions. Appl. Math. Comput. 283, 299–315 (2016)
Nishizawa, Y.: Sharpening of Jordan’s type and Shafer–Fink’s type inequalities with exponenti alapproximations. Appl. Math. Comput. 269, 146–154 (2015)
Nishizawa, Y.: Sharp exponential approximate inequalities for trigonometric functions. Results Math. 71(3–4), 609–621 (2017)
Qi, F.: Extensions and sharpenings of Jordan’s and Kober’s inequality. J. Math. Technol. 12(4), 98–102 (1996)
Rahmatollahi, G., De Abreu, G.T.F.: Closed-form hop-count distributions in random networks with arbitrary routing. IEEE Trans. Commun. 60(2), 429–444 (2012)
Rašajski, M., Lutovac, T., Malešević, B.: Sharpening and generalizations of Shafer–Fink and Wilker type inequalities: a new approach. J. Nonlinear Sci. Appl. 11(7), 885–893 (2018)
Wu, S., Debnath, L.: A generalization of L’Hospital-type rules for monotonicity and its application. Appl. Math. Lett. 22, 284–290 (2009)
Zhu, L.: Sharpening of Jordan’s inequalities and its applications. Math. Inequal. Appl. 9(1), 103–106 (2006)
Zhu, L.: Sharpening Jordan’s inequality and Yang Le’s inequality II. Appl. Math. Lett. 19(9), 990–994 (2006)
Compliance with ethical standards
Conflict of interest
The authors would like to state that they do not have any competing interests in the subject of this research.
Author information
Authors and Affiliations
Contributions
All the authors participated in every phase of the research conducted for this paper.
Corresponding author
Additional information
Research of the first and second and third author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under Projects TR 32023, ON 174032 & III 44006 and ON 174033, respectively.
Rights and permissions
About this article
Cite this article
Lutovac, T., Malešević, B. & Rašajski, M. A New Method for Proving Some Inequalities Related to Several Special Functions. Results Math 73, 100 (2018). https://doi.org/10.1007/s00025-018-0862-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0862-1