Computational Methods and Function Theory

, Volume 5, Issue 2, pp 373–379 | Cite as

Landen-Type Inequality for Bessel Functions

  • Árpád Baricz


Let u p(x) be the generalized and normalized Bessel function depending on parameters b,c,p and let λ(r) = u p(r2), r ∈} (0,1). Motivated by an open problem of Anderson, Vamanamurthy and Vuorinen, we prove that the Landen-type inequality λ(2√r/(1 + r)) < (r) holds for all r ∈ (0,1) and C > 1, for certain conditions on the parameters b,c,p.


Landen inequality hypergeometric functions Bessel functions Kummer functions 

2000 MSC

33C05 33C10 33C15 


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  1. 1.
    G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly 95 (1988), 585–608.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Hypergeometric functions and elliptic integrals, in: Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (eds.), pp. 48–85, World Scientific, Singapore/London, 1992.CrossRefGoogle Scholar
  3. 3.
    Á. Baricz, Geometric properties of generalized Bessel functions, manuscript.Google Scholar
  4. 4.
    S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), 43–64.MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. L. Qiu and M. Vuorinen, Landen inequalities for hypergeometric functions, Nagoya Math. J. 154 (1999), 31–56.MathSciNetMATHGoogle Scholar
  6. 6.
    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1962.Google Scholar

Copyright information

© Heldermann  Verlag 2005

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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