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′.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
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
Mitrinović D.S., Pečarić J.E., Fink A.M.: Classical and new inequalities in analysis. Kluwer Academic Publishers, New York (1993)
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)
Mortici, C.: Estimating gamma function by digamma function, Math. Com. Modell. 52(5), 942–946
Mortici, C.: Improved asymptotic formulas for the gamma function. Comp. Math. Appl. 61(11), 3364–3369
Mortici C.: Ramanujan formula for the generalized Stirling approximation. Appl. Math. Comp. 217(6), 2579–2585 (2010)
Mortici C.: New approximation formulas for evaluating the ratio of gamma functions. Math. Comp. Modell. 52(1–2), 425–433 (2010)
Mortici, C., Qi, F.: Asymptotic formulas and inequalities for gamma function in terms of tri-gamma function. http://arxiv.org/abs/1312.5881
Qi, F.: A completely monotonic function involving the gamma and tri-gamma functions. http://arxiv.org/abs/1307.5407
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
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
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)
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
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
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
Widder D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-015-0439-1