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Error-functions in double-sided Taylor’s approximations. (English) Zbl 1474.26056

Summary: In this paper we introduce the error-functions for one-sided and double-sided Taylor’s approximations of real analytic functions. We illustrate the application of error-functions in the process of generalization of one trigonometric inequality.

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26D15 Inequalities for sums, series and integrals
42A10 Trigonometric approximation
Full Text: DOI

References:

[1] H. Cox,: A demonstration of Taylor’s theorem, Cambridge and Dublin Math. J., 6, (1851) 8081.
[2] D. S. Mitrinovi¢,: Analytic inequalities, Springer (1970). · Zbl 0199.38101
[3] Ch.-P. Chen, F. Qi,: A double inequality for remainder of power series of tangent function, Tamkang J. Math. 34:3 (2003), 351355.
[4] S.-H. Wu, H.M. Srivastva: A further renement of a Jordan type inequality and its applications, Appl. Math. Comput. 197 (2008), 914923.
[5] S.-H. Wu, L. Debnath: Jordan-type inequalities for dierentiable functions and their applications, Appl. Math. Lett. 21:8 (2008), 803809.
[6] S.-H. Wu, H. M. Srivastava: A further renement of Wilker’s inequality, Integral Transforms Spec. Funct. 19:9-10 (2008), 757765.
[7] S. Wu, L. Debnath: A generalization of L’Hospital-type rules for monotonicity and its application, Appl. Math. Lett. 22:2 (2009), 284290.
[8] L. Zhu, J. Hua: Sharpening the Becker-Stark inequalities, J. Inequal. Appl. 2010 (2010), 14. · Zbl 1185.26059
[9] C. Mortici: The natural approach of Wilker-Cusa-Huygens inequalities, Math. In-equal. Appl. 14:3 (2011), 535541. · Zbl 1222.26020
[10] J.-L. Zhao, Q.-M. Luo, B.-N. Guo, and F. Qi: Remarks on inequalities for the tangent function, Hacettepe J. Math. Statist. 41:4 (2012), 499506.
[11] G. Milovanovi¢, M. Rassias (ed.): Analytic Number Theory, Approximation The-ory and Special Functions, Springer 2014. Chapter: G. D. Anderson, M. Vuorinen, X. Zhang: Topics in Special Functions III, 297345. · Zbl 1286.00055
[12] I.S. Gradshteyn, I.M Ryzhik: Table of Integrals Series and Products, 8-th edn. Academic Press, San Diego (2015)
[13] B. Banjac, M. Nenezi¢, B. Male²evi¢: Some applications of Lambda-method for obtaining approximations in lter design, Proceedings of 23-rd TELFOR conference, pp. 404-406, Beograd 2015.
[14] L. Debnath, C. Mortici, L. Zhu: Renements of Jordan-Steckin and Becker-Stark inequalities, Results Math. 67 (1-2) (2015), 207215.
[15] B. Male²evi¢, M. Makragi¢: A Method for Proving Some Inequalities on Mixed Trigonometric Polynomial Functions, J. Math. Inequal. 10:3 (2016), 849876.
[16] M. Nenezi¢, B. Male²evi¢, C. Mortici: New approximations of some expressions involving trigonometric functions, Appl. Math. Comput. 283 (2016), 299315.
[17] B. Banjac, M. Makragi¢, B. Male²evi¢: Some notes on a method for proving inequalities by computer, Results Math. 69:1 (2016), 161176.
[18] J. Sàndor: On D’aurizio’s trigonometric inequality, J. Math. Inequal. 10:3 (2016), 885888. · Zbl 1349.26035
[19] L. E. Persson, H. Rafeiro, P. Wall: Historical synopsis of the Taylor remainder, Note Mat. 37:1 (2017), 121. · Zbl 1387.26001
[20] M. Makragi¢: A method for proving some inequalities on mixed hyperbolic-trigonometric polynomial functions, J. Math. Inequal. 11:3 (2017), 817829.
[21] T. Lutovac, B. Male²evi¢, C. Mortici: The natural algorithmic approach of mixed trigonometric-polynomial problems, J. Inequal. Appl. 2017:116 (2017), 116. · Zbl 1373.42003
[22] B. Male²evi¢, M. Ra²ajski, T. Lutovac: Renements and generalizations of some inequalities of Shafer-Fink’s type for the inverse sine function, J. Inequal. Appl. 2017:275 (2017), 19. · Zbl 1374.26034
[23] B. Male²evi¢, I. Jovovi¢, B. Banjac: A proof of two conjectures of Chao-Ping Chen for inverse trigonometric functions, J. Math. Inequal. 11 (1) (2017), 151162.
[24] H. Alzer, M. K. Kwong: On Jordan’s inequality, Period. Math. Hung. 77:2 (2018), 191200. · Zbl 1413.26031
[25] B. Male²evi¢, T. Lutovac, M. Ra²ajski, C. Mortici: Extensions of the natural approach to renements and generalizations of some trigonometric inequalities, Adv. Dierence Equ. 2018:90 (2018), 115.
[26] M. Ra²ajski, T. Lutovac, B. Male²evi¢: Sharpening and generalizations of Shafer-Fink and Wilker type inequalities: a new approach, J. Nonlinear Sci. Appl. 11:7 (2018), 885893.
[27] M. Ra²ajski, T. Lutovac, B. Male²evi¢: About some exponential inequalities related to the sinc function, J. Inequal. Appl. 2018:150 (2018), 110. · Zbl 1498.26029
[28] T. Lutovac, B. Male²evi¢, M. Ra²ajski: A new method for proving some in-equalities related to several special functions, Results Math. 73:100 (2018), 115. · Zbl 1400.33001
[29] M. Nenezi¢, L. Zhu: Some improvements of Jordan-Steckin and Becker-Stark in-equalities, Appl. Anal. Discrete Math. 12 (2018), 244256.
[30] B. Male²evi¢, T. Lutovac, B. Banjac: A proof of an open problem of Yusuke Nishizawa for a power-exponential function, J. Math. Inequal. 12:2 (2018), 473485.
[31] B. Male²evi¢, M. Ra²ajski, T. Lutovac: Rened estimates and generalizations of inequalities related to the arctangent function and Shafer’s inequality, Math. Probl. Eng. 2018 Article ID 4178629, 18. · Zbl 1427.26004
[32] B. Male²evi¢, T. Lutovac, B. Banjac: One method for proving some classes of exponential analytical inequalities, Filomat 32:20 (2018), 69216925. · Zbl 1499.26060
[33] B. Male²evi¢, M. Ra²ajski, T. Lutovac: Double-sided Taylor’s approximations and their applications in Theory of analytic inequalities, in Ed. Th. Rassias and D. · Zbl 1441.41009
[34] Andrica: Dierential and Integral Inequalities, Springer Optimization and Its Appli-cations, vol 151. pp. 569-582, Springer 2019. · Zbl 1431.26003
[35] B. Male²evi¢, T. Lutovac M. Ra²ajski, B. Banjac: Double-Sided Taylor’s Approximations and Their Applications in Theory of Trigonometric Inequalities, in
[36] Ed. M.Th. Rassias, A. Raigorodskii: Trigonometric Sums and their Applications, pp. 159-167, Springer 2020 · Zbl 1443.39001
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