Redheffer type bounds for Bessel and modified Bessel functions of the first kind

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Abstract

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.

Keywords

Bessel and modified Bessel functions of the first and second kind Zeros of Bessel functions Redheffer type inequalities Rayleigh sum of zeros of Bessel functions of the first kind 

Mathematics Subject Classification

33C10 

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References

  1. 1.
    Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Inequalities for quasiconformal mappings in space. Pac. J. Math. 160(1), 1–18 (1993)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baricz, Á.: Redheffer type inequality for Bessel functions. J. Inequal. Pure Appl. Math. 8(1), 11 (2007)MathSciNetMATHGoogle Scholar
  3. 3.
    Baricz, Á.: Jordan-type inequalities for generalized Bessel functions. J. Inequal. Pure Appl. Math. 9(2), 39 (2008)MathSciNetMATHGoogle Scholar
  4. 4.
    Baricz, Á.: Functional inequalities involving Bessel and modied Bessel functions of the first kind. Expo. Math. 26(3), 279–293 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Baricz, Á.: Bounds for modified Bessel functions of the first and second kinds. Proc. Edinb. Math. Soc. 53(3), 575–599 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Baricz, Á.: Bounds for Turánians of modified Bessel functions. Expo. Math. 33(2), 223–251 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baricz, Á., Wu, S.: Sharp exponential Redheffer-type inequalities for Bessel functions. Publ. Math. Debr. 74, 257–278 (2009)MathSciNetMATHGoogle Scholar
  8. 8.
    Ifantis, E.K., Siafarikas, P.D.: Inequalities involving Bessel and modied Bessel functions. J. Math. Anal. Appl. 147(1), 214–227 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ismail, M.E.H.: Bessel functions and the infinite divisibility of the Student $t$ distribution. Ann. Probab. 5(4), 582–585 (1977)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kishore, N.: The Rayleigh function. Proc. Am. Math. Soc. 14, 527–533 (1963)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mehrez, K.: Redheffer type inequalities for modified Bessel functions. Arab. J. Math. Sci. 22(1), 38–42 (2016)MathSciNetMATHGoogle Scholar
  12. 12.
    Muldoon, M.E.: Convexity properties of special functions and their zeros. In: Milovanovic, G.V. (ed.) Recent Progress in Inequalities. Mathematics and Its Applications, vol. 430, pp. 309–323. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
  13. 13.
    Ponnusamy, S., Vuorinen, M.: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 44, 278–301 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Qi, F., Niu, D.-W., Guo, B.-N.: Refinements, generalizations, and applications of Jordan’s inequality and related problems. J. Inequal. Appl. 52, 271923 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Skovgaard, H.: On inequalities of the Turán type. Math. Scand. 2, 65–73 (1954)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    van de Lune, J.: Some inequalities involving Riemann’s zeta-function. Mathematisch Centrum, Afdeling Zuivere Wiskunde, ZW 50/75, Amsterdam (1975)Google Scholar
  17. 17.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1944)MATHGoogle Scholar
  18. 18.
    Zhu, L.: Extension of Redheffer type inequalities to modified Bessel functions. Appl. Math. Comput. 217, 8504–8506 (2011)MathSciNetMATHGoogle Scholar
  19. 19.
    Zhu, L., Sun, J.: Six new Redheffer-type inequalities for circular and hyperbolic functions. Comput. Math. Appl. 56(2), 522–529 (2008)MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of EconomicsBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.Institute of Applied MathematicsÓbuda UniversityBudapestHungary
  3. 3.Faculty of Sciences of TunisUniversity of Tunis El ManarTunisTunisia

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