, Volume 22, Issue 5, pp 1403–1417 | Cite as

A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

  • Jamel El Kamel
  • Khaled MehrezEmail author


In this paper, we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite. We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function.


Bessel functions Positive definite functions Completely monotonic functions Logarithmically completely monotonic functions 

Mathematics Subject Classification

42A82 33C10 26D07 


  1. 1.
    Bekner, W.: Inequalities in Fourier analysis. Ann. Math. (2) 102, 159–182 (1975)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bell, W.W.: Special Functions for Scientists and Engineers, London 1967. Encyclopedia of Mathematics and Its Application, vol. 35. Cambridge Univ. Press, Cambridge (1990)Google Scholar
  3. 3.
    Bochner, S.: Integral Transforms and Their Applications, Applied Math. Sciences, vol. 25. Springer, New YorkGoogle Scholar
  4. 4.
    Debnath, L.: Integral Transform and Their Applications. CRC Press, Inc., Boca Raton (1995)zbMATHGoogle Scholar
  5. 5.
    Derrien, F.: Strictly positive definite functionsd on the real line, hal-00519325, version 1-20sep (2010)Google Scholar
  6. 6.
    Fan, M.K.: Les fonctions définies positives et les fonctions complètement monotones. Memorial Sciences Mathématiques Paris, Paris (1950)zbMATHGoogle Scholar
  7. 7.
    Fitouhi, A.: Inégalité de Babenko et inégalité logarithmique de Sobolev pour l’opérateur de Bessel. C. R. Acad. Sci. Paris 305(I), 877–880 (1987)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Guo, S., Srivastava, H.M.: A certain function class related to the class of logarithmically completely monotonic functions. Math. Comput. Model. 49, 2073–2079 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ismail, M.E.H.: Bessel functions and the infinite divisibility of the student t-distribution. Ann. Probab. 5, 582–585 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ismail, M.E.H.: Complete monotonicity of the modified Bessel functions. Proc. Am. Math. Soc. 108(2), 353–361 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Schoenberg, I.J.: Metric spaces and completely monotonic functions. Ann. Math. 39, 811–841 (1938)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  13. 13.
    Wendland, H.: Scattered Data Approximations. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de MathématiquesFSMMonastirTunisia
  2. 2.Département de Mathématiques, Faculté de Sciences de TunisUniversité Tunis El ManarTunisTunisia
  3. 3.Département de Mathématiques ISSAT KasserineUniversité de KairouanKairouanTunisia

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