Refinements of Huygens-Wilker-Lazarović inequalities via the hyperbolic cosine polynomials. (English) Zbl 1499.42040
Summary: The aim of this paper is to provide new refinements of Huygens-Wilker-Lazarović inequalities using hyperbolic cosine polynomials. We give an unitary approach for both inequalities of trigonometric and hyperbolic functions.
MSC:
| 42B05 | Fourier series and coefficients in several variables |
| 26D05 | Inequalities for trigonometric functions and polynomials |
Keywords:
trigonometric functions; cosine polynomials; hyperbolic cosine polynomials; analytic inequalities; even functionsReferences:
| [1] | M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. In Appl. Math. Ser. National Bureau of Standards, 55, Washington, D. C., 1972. · Zbl 0543.33001 |
| [2] | B. Banjac, M. Makragić, B. Malešević: Some notes on a method for proving inequalities by computer. Results Math., 69 (1) (2016), 161-176. · Zbl 1332.41008 |
| [3] | G. Bercu: Padé approximant related to remarkable inequalities involving trigonometric functions. J. Inequal. Appl., 2016 (1), 99. · Zbl 1333.41006 |
| [4] | G. Bercu: Fourier Series Method Related to Wilker -Cusa -Huygens Inequalities. Math. Inequal. Appl., 22 (4) (2019), 1091-1098. · Zbl 1434.42007 |
| [5] | J. P. Boyd: Chebyshev and Fourier Spectral Methods. New York, USA: Dover, 2000. |
| [6] | C. P. Chen, W. S. Cheung: Wilker -and Huygens -type inequalities and solution to Oppenheim’s problem. Int. Trans. Spec. Funct., 23 (5) (2012), 325-336. · Zbl 1254.26022 |
| [7] | B. N. Guo, B. M. Qiao, F. Qi, W. Li: On new proofs of Wilker inequalities involving trigonometric functions. Math. Inequal. Appl., 6 (2003), 19-22. · Zbl 1040.26006 |
| [8] | T. Lutovac, B. Malešević, C. Mortici: The natural algorithmic approach of mixed trigonometric polynomial problems. J. Inequal. Appl., 2017 (2017), 116. · Zbl 1373.42003 |
| [9] | B. Malešević, B. Mihailović: A minimax approximant in the theory of analytic inequalities. Appl. Anal. Discrete Math., 15 (2021), 486-509. · Zbl 1499.26061 |
| [10] | B. Malešević, M. Rašajski, T. Lutovac: Double-sided Taylor’s approximations and their applications in Theory of analytic inequalities. In Ed. M. Th. Rassias and D. Andrica: Differential and Integral Inequalities, Springer Optimization and Its Appli-cations, 151, 569 -582, Springer, 2019. · Zbl 1441.41009 |
| [11] | B. Malešević, T. Lutovac, M. Rašajski, B. Banjac: Double-sided Taylor’s ap-proximations and Their Applications in Theory of Trigonometric Inequalities. In Ed. |
| [12] | M. Th. Rassias, A. Raigorodskii: Trigonometric Sums and their Applications, 159 -167, Springer, 2020. · Zbl 1443.39001 |
| [13] | B. Malešević, T. Lutovac, M. Rašajski, B. Banjac: Error-Functions in Double-sided Taylor’s Approximations. Appl. Anal. Discrete Math., 14(3) (2020), 599-613. · Zbl 1474.26056 |
| [14] | D. S. Mitrinović: Analytic Inequalities. Springer -Verlag, 1970. · Zbl 0199.38101 |
| [15] | C. Mortici: The natural approach of Wilker -Cusa -Huygens Inequalities. Math. Inequal. Appl., 14 (3) (2011), 535-541. · Zbl 1222.26020 |
| [16] | M. Nenezić, B. Malešević, C. Mortici: Accurate approximations of some expres-sions involving trigonometric functions. Appl. Math. Comput., 283 (2016), 299-315. · Zbl 1410.26025 |
| [17] | E. Neuman: Wilker and Huygens -type inequalities for Jacobian elliptic and theta functions. Int. Trans. Spec. Funct., 25 (3) (2014), 240-248. · Zbl 1283.26007 |
| [18] | E. Neuman, J. Sándor: On some inequalities involving trigonometric and hyperbolic functions with emphasis of the Cusa -Huygens, Wilker and Huygens inequalities. Math. Inequal. Appl., 13 (4) (2010), 715-723. · Zbl 1204.26023 |
| [19] | J. Wei -Dong, L. Qiu -Ming, Q. Teng: Refinements and sharpening of some Huygens and Wilker type inequalities. Turk. J. Anal. Number Theory, 2 (4) (2014), 134-139. |
| [20] | S. -H. Wu, A. Baricz: Generalizations of Mitrinovic, Adamovic and Lazarovic’s inequalities and their applications. Publ. Math. Debrecen, 75 (3-4) (2009), 447-458. · Zbl 1212.26032 |
| [21] | S. -H. Wu, H. -M. Srivastava: A further refinement of Wilker’s inequality. Int. Trans. Spec. Funct., 19 (10) (2008), 757-765. · Zbl 1176.11008 |
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