Advertisement

The Ramanujan Journal

, Volume 31, Issue 3, pp 301–313 | Cite as

Sharp bounds for the Landau constants

  • Chao-Ping Chen
Article

Abstract

The Landau constants are defined for all integers n≥0 by
$$ G_{n}=\sum_{k=0}^{n} \frac{1}{16^k}\binom{2k}{k}^2. $$
We establish sharp bounds for the Landau constants G n . Relevant connections of the results presented here with those derived in earlier work are also pointed out.

Keywords

Landau constants Gamma function Psi function Inequality Asymptotic expansion 

Mathematics Subject Classification (2010)

41A60 26D15 

Notes

Acknowledgements

The numerical calculations presented in this work were performed by using the Maple software for symbolic computations.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (1972). Ninth printing MATHGoogle Scholar
  2. 2.
    Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Alzer, H.: Inequalities for the constants of Landau and Lebesgue. J. Comput. Appl. Math. 139, 215–230 (2002) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Berndt, B.C.: Ramanujan’s Notebooks. Part II. Springer, Berlin (1999) Google Scholar
  5. 5.
    Brutman, L.: A sharp estimate of the Landau constants. J. Approx. Theory 34, 217–220 (1982) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Chen, C.-P.: Approximation formulas for Landau’s constants. J. Math. Anal. Appl. 387, 916–919 (2012) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cvijović, D., Klinowski, J.: Inequalities for the Landau constants. Math. Slovaca 50, 159–164 (2000) MathSciNetMATHGoogle Scholar
  8. 8.
    Falaleev, L.P.: Inequalities for the Landau constants. Sib. Math. J. 32, 896–897 (1991) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Granath, H.: On inequalities and asymptotic expansions for the Landau constants. J. Math. Anal. Appl. 386, 738–743 (2012) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kuang, J.-C.: Applied Inequalities, 3rd edn. Shandong Science and Technology Press, Jinan City (2004) (Chinese) Google Scholar
  11. 11.
    Landau, E.: Abschätzung der Koeffzientensumme einer Potenzreihe. Arch. Math. Phys. 21, 42–50, 250–255 (1913) MATHGoogle Scholar
  12. 12.
    Mortici, C.: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 23, 97–100 (2010) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mortici, C.: Product approximations via asymptotic integration. Am. Math. Mon. 117, 434–441 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mortici, C.: Sharp bounds of the Landau constants. Math. Comput. 80, 1011–1018 (2011) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Nemes, G., Nemes, A.: A note on the Landau constants. Appl. Math. Comput. 217, 8543–8546 (2011) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Popa, E.C.: Note of the constants of Landau. Gen. Math. 18, 113–117 (2010) MathSciNetMATHGoogle Scholar
  17. 17.
    Popa, E.C., Secelean, N.-A.: On some inequality for the Landau constants. Taiwan. J. Math. 15, 1457–1462 (2011) MathSciNetMATHGoogle Scholar
  18. 18.
    Wallis, J.: Arithmetica Infinitorum. Oxford University Press, London (1656). Facsimile of relevant pages available in: J.A. Stedall, Catching Proteus: The Collaborations of Wallis and Brouncker. I. Squaring the circle. Notes Rec. R. Soc. Lond. 54(3), 293–316 (2000) Google Scholar
  19. 19.
    Watson, G.N.: The constants of Landau and Lebesgue. Quart. J. Math. Oxford Ser. 1, 310–318 (1930) MATHCrossRefGoogle Scholar
  20. 20.
    Zhao, D.: Some sharp estimates of the constants of Landau and Lebesque. J. Math. Anal. Appl. 349, 68–73 (2009) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuo CityChina

Personalised recommendations