Sharpening and generalizations of Shafer-Fink and Wilker type inequalities: a new approach. (English) Zbl 1438.33002
Summary: In this paper, we propose and prove some generalizations and sharpenings of certain inequalities of Wilker’s and Shafer-Fink’s type. Application of the Wu-Debnath theorem enabled us to prove some double sided inequalities.
MSC:
| 33B10 | Exponential and trigonometric functions |
| 26D05 | Inequalities for trigonometric functions and polynomials |
References:
| [1] | 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, USA, 2884-2888 (2014) · doi:10.1109/ISIT.2014.6875361 |
| [2] | Anderson, D. G.; Vuorinen, M.; Zhang, X., Topics in Special Functions III, Analytic number theory, approximation theory and special functions, 297-345, Springer, New York (2014) · Zbl 1323.33001 |
| [3] | Bercu, G., Sharp Refinements for the Inverse Sine Function Related to Shafer-Fink’s Inequality, Math. Probl. Eng., 2017, 1-5 (2017) · Zbl 1426.26031 · doi:10.1155/2017/9237932 |
| [4] | Cloud, M. J.; Drachman, B. C.; Lebedev, L. P., Inequalities With Applications to Engineering, Springer, London (2014) · Zbl 1295.26002 |
| [5] | Abreu, G. T. F. De, Jensen-Cotes upper and lower bounds on the Gaussian Q-function and related functions, IEEE Trans. Commun., 57, 3328-3338 (2009) · doi:10.1109/TCOMM.2009.11.080479 |
| [6] | A. M. Fink, Two inequalities, Univ. Beograd Publ. Elektroteh. Fak. Ser. Mat., 6, 49-50 (1995) · Zbl 0841.26009 |
| [7] | Lutovac, T.; Malešević, B.; Mortici, C., The natural algorithmic approach of mixed trigonometric-polynomial problems, J. Inequal. Appl., 2017, 1-16 (2017) · Zbl 1373.42003 · doi:10.1186/s13660-017-1392-1 |
| [8] | Makragić, M., A method for proving some inequalities on mixed hyperbolic-trigonometric polynomial functions, J. Math. Inequal., 11, 817-829 (2017) · Zbl 1373.26017 · doi:10.7153/jmi-2017-11-63 |
| [9] | Malešević, B.; Makragić, M., A Method for Proving Some Inequalities on Mixed Trigonometric Polynomial Functions, J. Math. Inequal., 10, 849-876 (2016) · Zbl 1351.26030 · doi:10.7153/jmi-10-69 |
| [10] | Malešević, B.; Lutovac, T.; Rašajski, M.; C. Mortici, Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities, Adv. Difference Equ., 2018, 1-15 (2018) · Zbl 1445.26015 · doi:10.1186/s13662-018-1545-7 |
| [11] | 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, 1-9 (2017) · Zbl 1445.26015 · doi:10.1186/s13662-018-1545-7 |
| [12] | Malešević, B.; Rašajski, M.; Lutovac, T., Refined estimates and generalizations of inequalities related to the arctangent function and Shafer’s inequality, arXiv, 2017, 1-15 (2017) |
| [13] | Mitrinović, D. S., Analytic Inequalities, Springer-Verlag, New York-Berlin (1970) · Zbl 0199.38101 |
| [14] | C. Mortici, A subtly analysis of Wilker inequation, Appl. Math. Comput., 231, 516-520 (2014) · Zbl 1410.26024 · doi:10.1016/j.amc.2014.01.017 |
| [15] | Nenezić, M.; Malešević, B.; Mortici, C., New approximations of some expressions involving trigonometric functions, Appl. Math. Comput., 283, 299-315 (2016) · Zbl 1410.26025 · doi:10.1016/j.amc.2016.02.035 |
| [16] | Rahmatollahi, G.; Abreu, G. T. F. De, Closed-Form Hop-Count Distributions in Random Networks with Arbitrary Routing, IEEE Trans. Commun., 60, 429-444 (2012) · doi:10.1109/TCOMM.2012.010512.110125 |
| [17] | Shafer, R. E., Elementary Problems 1867, Amer. Math. Monthly, 73, 1-309 (1966) · doi:10.2307/2315358 |
| [18] | J. B. Wilker, Elementary Problems 3306, Amer. Math. Monthly, 96, 1-55 (1989) · doi:10.2307/2323259 |
| [19] | Wu, S.; Debnath, L., A generalization of L’Hospital-type rules for monotonicity and its application, Appl. Math. Lett., 22, 284-290 (2009) · Zbl 1163.26312 · doi:10.1016/j.aml.2008.06.001 |
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