Skip to main content
  • Research Article
  • Open access
  • Published:

On some Turán-type inequalities


We prove Turán-type inequalities for some special functions by using a generalization of the Schwarz inequality.



  1. Csordas G, Norfolk TS, Varga RS: The Riemann hypothesis and the Turán inequalities. Transactions of the American Mathematical Society 1986,296(2):521–541.

    MATH  MathSciNet  Google Scholar 

  2. Elbert Á, Laforgia A: Some monotonicity properties of the zeros of ultraspherical polynomials. Acta Mathematica Hungarica 1986,48(1–2):155–159. 10.1007/BF01949060

    Article  MATH  MathSciNet  Google Scholar 

  3. Elbert Á, Laforgia A: Monotonicity results on the zeros of generalized Laguerre polynomials. Journal of Approximation Theory 1987,51(2):168–174. 10.1016/0021-9045(87)90031-1

    Article  MATH  MathSciNet  Google Scholar 

  4. Gradshteyn IS, Ryzhik IM: Table of Integrals, Series, and Products. 6th edition. Academic Press, California; 2000:xlvii+1163.

    MATH  Google Scholar 

  5. Katkova OM: Multiple positivity and the Riemann zeta-function.

  6. Laforgia A: Sturm theory for certain classes of Sturm-Liouville equations and Turánians and Wronskians for the zeros of derivative of Bessel functions. Indagationes Mathematicae 1982,44(3):295–301.

    Article  MATH  MathSciNet  Google Scholar 

  7. Laforgia A, Natalini P: Turán-type inequalities for some special functions. submitted submitted

  8. Lorch L: Turánians and Wronskians for the zeros of Bessel functions. SIAM Journal on Mathematical Analysis 1980,11(2):223–227. 10.1137/0511021

    Article  MATH  MathSciNet  Google Scholar 

  9. Pólya G: Collected Papers. Vol. II: Location of Zeros, edited by R. P. Boas, Mathematicians of Our Time. Volume 8. The MIT Press, Massachusetts; 1974.

    Google Scholar 

  10. Szegö G: Orthogonal Polynomials, Colloquium Publications. Volume 23. 4th edition. American Mathematical Society, Rhode Island; 1975:xiii+432.

    Google Scholar 

  11. Titchmarsh EC: The Theory of the Riemann Zeta-Function. The Clarendon Press, Oxford; 1951:vi+346.

    MATH  Google Scholar 

  12. Turán P: On the zeros of the polynomials of Legendre. Časopis Pro Pěstování Matematiky 1950, 75: 113–122.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. Laforgia.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Laforgia, A., Natalini, P. On some Turán-type inequalities. J Inequal Appl 2006, 29828 (2006).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: