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Journal of Global Optimization

, Volume 57, Issue 4, pp 1173–1192 | Cite as

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

  • Gradimir V. Milovanović
  • Michael Th. Rassias
Article

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.

Keywords

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

Mathematics Subject Classification (2000)

26D07 26D15 33C05 41A10 41A17 49K05 

References

  1. 1.
    Alpár, L.: In memory of Paul Turán. J. Number Theory 13, 271–278 (1981)CrossRefGoogle Scholar
  2. 2.
    Alzer, H., Felder, G.: A Turán-type inequality for the gamma function. J. Math. Anal. Appl. 350, 276–282 (2009)CrossRefGoogle Scholar
  3. 3.
    Alzer, H., Gerhold, S., Kauers, M., Lupaş, A.: On Turán’s inequality for Legendre polynomials. Expo. Math. 25, 181–186 (2007)CrossRefGoogle Scholar
  4. 4.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, New York (1999)CrossRefGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. 6.
    Baricz, Á.: Turán type inequalities for generalized complete elliptic integrals. Math. Z. 256, 895–911 (2007)Google Scholar
  7. 7.
    Baricz, Á.: Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expo. Math. 26, 279–293 (2008)Google Scholar
  8. 8.
    Baricz, Á.: Turán type inequalities for hypergeometric functions. Proc. Am. Math. Soc. 136, 3223–3229 (2008)Google Scholar
  9. 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. 10.
    Dimitrov, D.K., Kostov, V.P.: Sharp Turán inequalities via very hyperbolic polynomials. J. Math. Anal. Appl. 376, 385–392 (2011)CrossRefGoogle Scholar
  11. 11.
    Ismail, M.E.H., Laforgia, A.: Monotonicity properties of determinants of special functions. Constr. Approx. 26, 1–9 (2007)CrossRefGoogle Scholar
  12. 12.
    Luke, Y.L.: Mathematical Functions and Their Approximations. Academic Press, New York (1975)Google Scholar
  13. 13.
    Mastroianni, G., Milovanović, G.V.: Interpolation Processes—Basic Theory and Applications. Springer verlag, Berlin (2008)CrossRefGoogle Scholar
  14. 14.
    Mastroianni, G., Milovanović, G.V., Notarangelo, I.: Gaussian quadrature rules with an exponential weight on the real semiaxis, (in preparation)Google Scholar
  15. 15.
    McEliece, R.J., Reznick, B., Shearer, J.B.: A Turán inequality arising in information theory. SIAM J. Math. Anal. 12, 931–934 (1981)CrossRefGoogle Scholar
  16. 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)Google Scholar
  17. 17.
    Sprugnoli, R.: Riordan Array Proofs of Identities in Gould’s Book, Preprint, Firenze (2006). http://www.dsi.unifi.it/resp/GouldBK.pdf
  18. 18.
    Szegő, G.: On an inequality of P. Turán concerning Legendre polynomials. Bull. Am. Math. Soc. 54, 401–405 (1948)CrossRefGoogle Scholar
  19. 19.
    Turán, P.: On the zeros of the polynomials of Legendre. Čas. Pěst. Mat. Fys. 75, 113–122 (1950)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Gradimir V. Milovanović
    • 1
  • Michael Th. Rassias
    • 2
  1. 1.Mathematical InstituteSerbian Academy of Science and ArtsBelgradeSerbia
  2. 2.Department of MathematicsETH-ZentrumZurichSwitzerland

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