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New refinements of Becker-Stark and Cusa-Huygens inequalities via trigonometric polynomials method

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The aim of this paper is to provide new refinements of Becker-Stark inequality and Cusa-Huygens inequality using trigonometric polynomial method. It is shown that the approach proposed is useful for establishing new inequalities and refining some existing inequalities.

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References

  1. Becker, M., Stark, E.L.: On a hierarchy of quolynomial inequalities for tan x. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 602–633, 133–138 (1978)

    MathSciNet  MATH  Google Scholar 

  2. Bercu, G.: The natural approach of trigonometric inequalities-Padé approximant. J. Math. Inequal. 11(1), 181–191 (2017)

    Article  MathSciNet  Google Scholar 

  3. Bercu, G.: Padé approximant related to remarkable inequalities involving trigonometric functions. J. Inequal. Appl. 2016, 11 (2016)

    Article  Google Scholar 

  4. Huygens, C.: Oeuvres Complètes, Publiées Par la Société hollandaise des sciences. Haga 20 (1888)

  5. Mortici, C.: The natural approach of Wilker-Cusa-Huygens inequalities. Math. Inequal. Appl. 14(3), 535–541 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Mitrinović, D.S., Adamović, D.D.: Sur une iné galité élémentaire où interviennent des fonctions trigonom étriques. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 143–155, 23–34 (1965)

    MATH  Google Scholar 

  7. Mitrinović, D.S.: Analytic Inequalities. Springer, Berlin (1970)

    Book  Google Scholar 

  8. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht (1993)

    Book  Google Scholar 

  9. Nenezić, M., Zhu, L.: Some improvements of Jordan-Stečkin and Becker-Stark Inequalities. Appl. Anal. Discrete Math. 12(1), 244–256 (2018)

    Article  MathSciNet  Google Scholar 

  10. Sun, Z.J., Zhu, L.: Simple proofs of the Cusa-Huygens-type and Becker-Stark-type inequalities. J. Math. Inequal. 7(4), 563–567 (2013)

    Article  MathSciNet  Google Scholar 

  11. Wu, S., Srivastava, H.M.: A weighted and exponential generalization of Wilker’s inequality and its applications. Integral Transform. Spec. Funct. 18(8), 529–535 (2007)

    Article  MathSciNet  Google Scholar 

  12. Wu, S., Srivastava, H.M.: A further refinement of a Jordan type inequality and its application. Appl. Math. Comput. 197, 914–923 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Wu, S., Srivastava, H.M., Debnath, L.: Some refined families of Jordan type inequalities and their applications. Integral Transform. Spec. Funct. 19(3), 183–193 (2008)

    Article  MathSciNet  Google Scholar 

  14. Wu, S., Debnath, L.: Wilker-type inequalities for hyperbolic functions. Appl. Math. Lett. 25(5), 837–842 (2012)

    Article  MathSciNet  Google Scholar 

  15. Wu, S., Baricz, Á.: Generalizations of Mitrinović, Adamović and Lazarević’s inequalities and their applications. Publ. Math. Debrecen 75(3–4), 447–458 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Zhu, L.: Sharp Becker-Stark-Type inequalities for Bessel functions. J. Inequal. Appl. 2010, 4 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Zhu, L.: A refinement of the Becker-Stark inequalities. Math. Notes 93(3–4), 421–425 (2013)

    Article  MathSciNet  Google Scholar 

  18. Zhu, L.: Sharp inequalities of Mitrinovíc-Adamovíc type. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(2), 957–968 (2019)

    Article  MathSciNet  Google Scholar 

  19. Zhu, L.: Some new bounds for Sinc function by simultaneous approximation of the base and exponential functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(2), 14 (2020)

  20. Zhu, L.: New Mitrinović-Adamović type inequalitie. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(3), 14 (2020)

    MATH  Google Scholar 

  21. Zhu, L.: An unity of Mitrinovíc-Adamovíc and Cusa-Huygens inequalities and the analogue for hyperbolic functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4), 3399–3412 (2019)

    Article  MathSciNet  Google Scholar 

  22. Zhu, L.: Cusa-Huygens type inequalities. AIMS Math. 5(5), 5320–5331 (2020)

    Article  MathSciNet  Google Scholar 

  23. Zhu, L., Sun, Z.J.: Refinements of Huygens and Wilker-type inequalities. AIMS Math. 5(4), 2967–2978 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Natural Science Foundation of Zhejiang Province (no. LY21A010016).

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Authors and Affiliations

  1. Department of Mathematics, China Jiliang University, Hangzhou, 310018, China

    Yiting Wu

  2. Department of Mathematics and Computer Sciences, “Dunărea de Jos” University of Galaţi, Galaţi, Romania

    Gabriel Bercu

Authors
  1. Yiting Wu
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  2. Gabriel Bercu
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All authors read and approved the final manuscript.

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Correspondence to Yiting Wu.

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Wu, Y., Bercu, G. New refinements of Becker-Stark and Cusa-Huygens inequalities via trigonometric polynomials method. RACSAM 115, 87 (2021). https://doi.org/10.1007/s13398-021-01030-6

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  • DOI: https://doi.org/10.1007/s13398-021-01030-6

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