Series representations of the remainders in the expansions for certain trigonometric functions and some related inequalities. II. (English) Zbl 1506.11035
Summary: We examine Wilker and Huygens-type inequalities involving trigonometric functions making use of results derived in Part I. The Papenfuss-Bach inequality representing upper and lower bounds for the function \(x \sec^2x - \tan x\) for \(0 \le x < \pi /2\) is also investigated. An open problem posed by Sun and Zhu concerning this last inequality is established.
For Part I, see [the authors, Math. Inequal. Appl. 20, No. 4, 1003–1016 (2017; Zbl 1441.11034)].
For Part I, see [the authors, Math. Inequal. Appl. 20, No. 4, 1003–1016 (2017; Zbl 1441.11034)].
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 26D05 | Inequalities for trigonometric functions and polynomials |
| 33B10 | Exponential and trigonometric functions |
Keywords:
Wilker inequality; Huygens inequality; Papenfuss-Bach inequality; trigonometric functions; inequalitiesCitations:
Zbl 1441.11034Software:
DLMFReferences:
| [1] | Abramowitz, M., Stegun, I.A. (Eds): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, vol. 55, Ninth printing, National Bureau of Standards, Washington, DC (1972) · Zbl 0543.33001 |
| [2] | Bach, G., Trigonometric inequality, Am. Math. Monthly, 87, 1, 62 (1980) |
| [3] | Baricz, A.; Sándor, J., Extensions of generalized Wilker inequality to Bessel functions, J. Math. Inequal., 2, 397-406 (2008) · Zbl 1171.33303 · doi:10.7153/jmi-02-35 |
| [4] | Chen, C-P; Cheung, W-S, Inequalities and solution to Oppenheim’s problem, Integral Transforms Spec. Funct., 23, 5, 325-336 (2012) · Zbl 1254.26022 · doi:10.1080/10652469.2011.586637 |
| [5] | Chen, C-P; Cheung, W-S, Sharpness of Wilker and Huygens type inequalities, J. Inequal. Appl., 2012, 72 (2012) · Zbl 1279.26030 · doi:10.1186/1029-242X-2012-72 |
| [6] | Chen, C-P; Paris, RB, Series representations of the remainders in the expansions for certain trigonometric functions and some related inequalities, I, Math. Inequal. Appl., 20, 4, 1003-1016 (2017) · Zbl 1441.11034 |
| [7] | Chen, C.-P., Paris, R.B.: Series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions with applications. http://arxiv.org/abs/1601.03180 |
| [8] | Chen, C-P; Paris, RB, On the Wilker and Huygens-type inequalities, J. Math. Inequalities, 14, 3, 685-705 (2020) · Zbl 1462.26014 · doi:10.7153/jmi-2020-14-44 |
| [9] | Chen, C-P; Qi, F., A double inequality for remainder of power series of tangent function, Tamkang J. Math., 34, 4, 351-355 (2003) · Zbl 1047.26012 · doi:10.5556/j.tkjm.34.2003.236 |
| [10] | Chen, C-P; Sándor, J., Inequality chains for Wilker, Huygens and Lazarević type inequalities, J. Math. Inequal., 8, 1, 55-67 (2014) · Zbl 1294.26016 · doi:10.7153/jmi-08-02 |
| [11] | Chen, X-D; Wang, H.; Yang, K.; Xie, J., New bounds of Wilker- and Huygens-type inequalities for inverse trigonometric functions, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM, 115, 1, 14 (2021) · Zbl 1473.26014 · doi:10.1007/s13398-020-00969-2 |
| [12] | Ge, H-F, New sharp bounds for the Bernoulli numbers and refinement of Becker-Stark inequalities, J. Appl. Math., 2012, 137507 (2012) · Zbl 1294.11016 · doi:10.1155/2012/137507 |
| [13] | Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press Inc., San Diego (2000). (Translated from the Russian. Sixth edition. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger.) · Zbl 0981.65001 |
| [14] | Guo, B-N; Qiao, B-M; Qi, F.; Li, W., On new proofs of Wilker inequalities involving trigonometric functions, Math. Inequal. Appl., 6, 19-22 (2003) · Zbl 1040.26006 |
| [15] | Huygens, C.: Oeuvres Completes 1888-1940, Société Hollondaise des Science, Haga |
| [16] | Mitrinović, DS, Analytic Inequalities (1970), Berlin: Springer-Verlag, Berlin · Zbl 0199.38101 · doi:10.1007/978-3-642-99970-3 |
| [17] | Mortici, C., The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal. Appl., 14, 535-541 (2011) · Zbl 1222.26020 |
| [18] | Mortici, C., A subtly analysis of Wilker inequality, Appl. Math. Comput., 231, 516-520 (2014) · Zbl 1410.26024 |
| [19] | Neuman, E., One- and two-sided inequalities for Jacobian elliptic functions and related results, Integral Transforms Spec. Funct., 21, 399-407 (2010) · Zbl 1193.33234 · doi:10.1080/10652460903345961 |
| [20] | Neuman, E., On Wilker and Huygens type inequalities, Math. Inequal. Appl., 15, 2, 271-279 (2012) · Zbl 1241.26011 |
| [21] | Neuman, E.; Sándor, J., On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math. Inequal. Appl., 13, 715-723 (2010) · Zbl 1204.26023 |
| [22] | Olver, FWJ; Lozier, DW; Boisvert, RF; Clarks, CW, NIST Handbook of Mathematical Functions (2010), New York: Cambridge University Press, New York · Zbl 1198.00002 |
| [23] | Papenfuss, MC, Problem E2739, Am. Math. Monthly, 85, 9, 765 (1978) · doi:10.2307/2321692 |
| [24] | Pinelis, I., L’Hospital rules of monotonicity and Wilker-Anglesio inequality, Am. Math. Monthly, 111, 905-909 (2004) · Zbl 1187.26010 · doi:10.1080/00029890.2004.11920156 |
| [25] | Sumner, JS; Jagers, AA; Vowe, M.; Anglesio, J., Inequalities involving trigonometric functions, Am. Math. Monthly, 98, 264-267 (1991) · doi:10.2307/2325035 |
| [26] | Sun, Z.; Zhu, L., Some refinements of inequalities for circular functions, J. Appl. Math., 2011, 869261 (2011) · Zbl 1235.26017 |
| [27] | Wilker, JB, Problem E 3306, Am. Math. Monthly, 96, 55 (1989) · doi:10.2307/2323260 |
| [28] | Wu, S-H, On extension and refinement of Wilker’s inequality, Rocky Mountain J. Math., 39, 683-687 (2009) · Zbl 1172.26008 |
| [29] | Wu, S-H; Baricz, A., Generalizations of Mitrinović, Adamović and Lazarevic’s inequalities and their applications, Publ. Math. Debrecen, 75, 447-458 (2009) · Zbl 1212.26032 |
| [30] | Wu, S-H; Srivastava, HM, A weighted and exponential generalization of Wilker’s inequality and its applications, Integral Transforms Spec. Funct., 18, 529-535 (2007) · Zbl 1128.26017 · doi:10.1080/10652460701284164 |
| [31] | Wu, S-H; Srivastava, HM, A further refinement of Wilker’s inequality, Integral Transforms Spec. Funct., 19, 757-765 (2008) · Zbl 1176.11008 · doi:10.1080/10652460802340931 |
| [32] | Zhang, L.; Zhu, L., A new elementary proof of Wilker’s inequalities, Math. Inequal. Appl., 11, 149-151 (2008) · Zbl 1138.26307 |
| [33] | Zhao, J-L; Luo, Q-M; Guo, B-N; Qi, F., Remarks on inequalities for the tangent function, Hacet. J. Math. Stat., 41, 4, 499-506 (2012) · Zbl 1273.26019 |
| [34] | Zhu, L., A new simple proof of Wilker’s inequality, Math. Inequal. Appl., 8, 2005, 749-750 (2005) · Zbl 1084.26008 |
| [35] | Zhu, L., Some new inequalities of the Huygens type, Comput. Math. Appl., 58, 1180-1182 (2009) · Zbl 1189.26030 · doi:10.1016/j.camwa.2009.07.045 |
| [36] | Zhu, L., Some new Wilker-type inequalities for circular and hyperbolic functions, Abstr. Appl. Anal., 2009, 485842 (2009) · Zbl 1177.33002 |
| [37] | Zhu, L., A source of inequalities for circular functions, Comput. Math. Appl., 58, 1998-2004 (2009) · Zbl 1189.26029 · doi:10.1016/j.camwa.2009.07.076 |
| [38] | Zhu, L., Inequalities for hyperbolic functions and their applications, J. Inequal. Appl., 2010, 130821 (2010) · Zbl 1221.26022 · doi:10.1155/2010/130821 |
| [39] | Zhu, L., Wilker inequalities of exponential type for circular functions, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM, 115, 1, 12 (2021) · Zbl 1468.26011 · doi:10.1007/s13398-020-00973-6 |
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