Skip to main content
SpringerLink
Log in

Asymptotic Formulas and Inequalities for the Gamma Function in Terms of the Tri-Gamma Function

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In the paper, the authors establish some asymptotic formulas and double inequalities for the factorial n! and the gamma function Γ in terms of the tri-gamma function ψ′.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I. A. (Eds): Handbook of mathematical functions with formulas, graphs, and mathematical tables, national bureau of standards, applied mathematics series vol. 55, 10th printing, Dover Publications, New York and Washington (1972)

  2. Koumandos S.: Remarks on some completely monotonic functions. J. Math. Anal. Appl. 324(2), 1458–1461 (2006). doi:10.1016/j.jmaa.2005.12.017

    Article  MATH  MathSciNet  Google Scholar 

  3. Mitrinović D.S., Pečarić J.E., Fink A.M.: Classical and new inequalities in analysis. Kluwer Academic Publishers, New York (1993)

    Book  MATH  Google Scholar 

  4. Mortici C.: A quicker convergence toward the γ constant with the logarithm term involving the constant e. Carpathian J. Math. 26(1), 86–91 (2010)

    MATH  MathSciNet  Google Scholar 

  5. Mortici, C.: Estimating gamma function by digamma function, Math. Com. Modell. 52(5), 942–946

  6. Mortici, C.: Improved asymptotic formulas for the gamma function. Comp. Math. Appl. 61(11), 3364–3369

  7. Mortici C.: Ramanujan formula for the generalized Stirling approximation. Appl. Math. Comp. 217(6), 2579–2585 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mortici C.: New approximation formulas for evaluating the ratio of gamma functions. Math. Comp. Modell. 52(1–2), 425–433 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mortici, C., Qi, F.: Asymptotic formulas and inequalities for gamma function in terms of tri-gamma function. http://arxiv.org/abs/1312.5881

  10. Qi, F.: A completely monotonic function involving the gamma and tri-gamma functions. http://arxiv.org/abs/1307.5407

  11. Qi, F.: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010 (2010), Article ID 493058, p 84, doi:10.1155/2010/493058

  12. Qi, F., Luo, Q.-M.: Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem. J. Inequal. Appl. 2013, 2013:542, 20. doi:10.1186/1029-242X-2013-542

  13. Qi F., Luo Q.-M.: Bounds for the ratio of two gamma functions: from Wendel’s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 6(2), 132–158 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Qi F., Luo Q.-M.: Complete monotonicity of a function involving the gamma function and applications. Period. Math. Hungar. 69(2), 159–169 (2014). doi:10.1007/s10998-014-0056-x

    Article  MATH  MathSciNet  Google Scholar 

  15. Qi F., Mortici C.: Some best approximation formulas and inequalities for the Wallis ratio. Appl. Math. Comp. 253, 363–368 (2015). doi:10.1016/j.amc.2014.12.039

    Article  MathSciNet  Google Scholar 

  16. Şevli H., Batır N.: Complete monotonicity results for some functions involving the gamma and polygamma functions. Math. Comput. Modell. 53, 1771–1775 (2011). doi:10.1016/j.mcm.2010.12.055

    Article  MATH  Google Scholar 

  17. Widder D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Valahia University of Târgovişte, Bd. Unirii 18, 130082, Târgovişte, Romania

    Cristinel Mortici

  2. Academy of the Romanian Scientists, Splaiul Independenţei 54, 050094, Bucharest, Romania

    Cristinel Mortici

  3. College of Mathematics, Inner Mongolia University for Nationalities, Inner Mongolia Autonomous Region, 028043, Tongliao City, China

    Feng Qi

  4. Department of Mathematics, College of Science, Tianjin Polytechnic University, 300387, Tianjin City, China

    Feng Qi

Authors
  1. Cristinel Mortici
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristinel Mortici.

Additional information

The work of the first author was supported by the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Grant No. PN-II-ID-PCE-2011-3-0087. The second author was partially supported by the National Natural Science Foundation of China under the Grant No. 11361038.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mortici, C., Qi, F. Asymptotic Formulas and Inequalities for the Gamma Function in Terms of the Tri-Gamma Function. Results. Math. 67, 395–402 (2015). https://doi.org/10.1007/s00025-015-0439-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-015-0439-1

Mathematics Subject Classification

Keywords

Search

Navigation