Skip to main content

Some properties of a hypergeometric function which appear in an approximation problem

Abstract

In this paper we consider properties and power expressions of the functions \(f:(-1,1)\rightarrow \mathbb{R }\) and \(f_L:(-1,1)\rightarrow \mathbb{R }\), defined by

$$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1-t^2}}\,\mathrm{d}t, \end{aligned}$$

respectively, where \(\gamma \) is a real parameter, as well as some properties of a two parametric real-valued function \(D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }\), defined by

$$\begin{aligned} D(x;\alpha ,\beta )= f(x;\beta )f(x;-\alpha -1)- f(x;-\alpha )f(x;\beta -1),\quad \alpha ,\beta \in \mathbb{R }. \end{aligned}$$

The inequality of Turán type

$$\begin{aligned} D(x;\alpha ,\beta )>0,\quad -1<x<1, \end{aligned}$$

for \(\alpha +\beta >0\) is proved, as well as an opposite inequality if \(\alpha +\beta <0\). Finally, for the partial derivatives of \(D(x;\alpha ,\beta )\) with respect to \(\alpha \) or \(\beta \), respectively \(A(x;\alpha ,\beta )\) and \(B(x;\alpha ,\beta )\), for which \(A(x;\alpha ,\beta )=B(x;-\beta ,-\alpha )\), some results are obtained. We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some “truncated” quadrature formulas of Gaussian type with an exponential weight on the real semiaxis, especially in a computation of Mhaskar–Rahmanov–Saff numbers.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Alpár, L.: In memory of Paul Turán. J. Number Theory 13, 271–278 (1981)

    Article  Google Scholar 

  2. Alzer, H., Felder, G.: A Turán-type inequality for the gamma function. J. Math. Anal. Appl. 350, 276–282 (2009)

    Article  Google Scholar 

  3. Alzer, H., Gerhold, S., Kauers, M., Lupaş, A.: On Turán’s inequality for Legendre polynomials. Expo. Math. 25, 181–186 (2007)

    Article  Google Scholar 

  4. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, New York (1999)

    Book  Google Scholar 

  5. Barnard, R.W., Gordy, M.B., Richards, K.C.: A note on Turán type and mean inequalities for the Kummer function. J. Math. Anal. Appl. 349, 259–263 (2009)

    Article  Google Scholar 

  6. Baricz, Á.: Turán type inequalities for generalized complete elliptic integrals. Math. Z. 256, 895–911 (2007)

  7. Baricz, Á.: Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expo. Math. 26, 279–293 (2008)

  8. Baricz, Á.: Turán type inequalities for hypergeometric functions. Proc. Am. Math. Soc. 136, 3223–3229 (2008)

  9. Baricz, Á., Jankov, D., Pogány, T.K.: Turán type inequalities for Krätzel functions. J. Math. Anal. Appl. 388, 716–724 (2012)

    Google Scholar 

  10. Dimitrov, D.K., Kostov, V.P.: Sharp Turán inequalities via very hyperbolic polynomials. J. Math. Anal. Appl. 376, 385–392 (2011)

    Article  Google Scholar 

  11. Ismail, M.E.H., Laforgia, A.: Monotonicity properties of determinants of special functions. Constr. Approx. 26, 1–9 (2007)

    Article  Google Scholar 

  12. Luke, Y.L.: Mathematical Functions and Their Approximations. Academic Press, New York (1975)

    Google Scholar 

  13. Mastroianni, G., Milovanović, G.V.: Interpolation Processes—Basic Theory and Applications. Springer verlag, Berlin (2008)

    Book  Google Scholar 

  14. Mastroianni, G., Milovanović, G.V., Notarangelo, I.: Gaussian quadrature rules with an exponential weight on the real semiaxis, (in preparation)

  15. McEliece, R.J., Reznick, B., Shearer, J.B.: A Turán inequality arising in information theory. SIAM J. Math. Anal. 12, 931–934 (1981)

    Article  Google Scholar 

  16. Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Vol. 1. Elementary functions (Translated from the Russian and with a preface by N. M. Queen). Gordon and Breach Science Publishers, New York (1986)

  17. Sprugnoli, R.: Riordan Array Proofs of Identities in Gould’s Book, Preprint, Firenze (2006). http://www.dsi.unifi.it/resp/GouldBK.pdf

  18. Szegő, G.: On an inequality of P. Turán concerning Legendre polynomials. Bull. Am. Math. Soc. 54, 401–405 (1948)

    Article  Google Scholar 

  19. Turán, P.: On the zeros of the polynomials of Legendre. Čas. Pěst. Mat. Fys. 75, 113–122 (1950)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gradimir V. Milovanović.

Additional information

The first author was supported in part by the Serbian Ministry of Education, Science and Technological Development.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Milovanović, G.V., Rassias, M.T. Some properties of a hypergeometric function which appear in an approximation problem. J Glob Optim 57, 1173–1192 (2013). https://doi.org/10.1007/s10898-012-0016-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-0016-z

Keywords

  • Approximation
  • Expansion
  • Minimum
  • Maximum
  • Turán type inequality
  • Hypergeometric function
  • Gamma function
  • Digamma function

Mathematics Subject Classification (2000)

  • 26D07
  • 26D15
  • 33C05
  • 41A10
  • 41A17
  • 49K05