Sharp inequalities for factorial n
DOI:
https://doi.org/10.4067/S0716-09172008000100006Keywords:
Factorial n, gamma function, Stirling’s formula, Burnside’s formula, función gamma, fórmula de Stirling, fórmula de Burnside.Abstract
Let n be a positive integer. We prove
with the best possible constants
This refines and extends a result of Sandor and Debnath, who proved that the double inequality holds with a = 0 and ß = 1.
References
[1] W. Burnside, A rapidly convergent series for logN!, Messenger Math., 46, pp. 157-159, (1917).
[2] S. Guo, Monotonicty and concavity properties of some functions involving the gamma function with applications, J. Inequal. Pure Appl. Math. 7, No. 2, article 45, (2006).
[3] M. Fichtenholz, Differential und integralrechnung II, Verlag Wiss, Berlin, (1978).
[4] E. A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comp. Appl. Math. 135.2, pp. 225-240, (2001).
[5] E. A. Karatsuba, On the computation of the Euler constant γ, Numerical Algorithms, 24, pp. 83-97, (2000).
[6] N. D. Mermin, Improving and improved analytical approximation to n!, Amer. J. Phys., 51, pp. 776, (1983).
[7] J. Sandor and L. Debnath, On certain inequalities involving the constant e and their applications, J. Math. Anal. Appl. 249, pp. 569-582, (2000).
[8] W. Schuster, Improving Stirling’s formula, Arch. Math. 77, pp. 170-176, (2001).
[9] Y. Weissman, An improved analytical approximation to n!, Amer. J. Phys., 51, No. 9, (1983).
[2] S. Guo, Monotonicty and concavity properties of some functions involving the gamma function with applications, J. Inequal. Pure Appl. Math. 7, No. 2, article 45, (2006).
[3] M. Fichtenholz, Differential und integralrechnung II, Verlag Wiss, Berlin, (1978).
[4] E. A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comp. Appl. Math. 135.2, pp. 225-240, (2001).
[5] E. A. Karatsuba, On the computation of the Euler constant γ, Numerical Algorithms, 24, pp. 83-97, (2000).
[6] N. D. Mermin, Improving and improved analytical approximation to n!, Amer. J. Phys., 51, pp. 776, (1983).
[7] J. Sandor and L. Debnath, On certain inequalities involving the constant e and their applications, J. Math. Anal. Appl. 249, pp. 569-582, (2000).
[8] W. Schuster, Improving Stirling’s formula, Arch. Math. 77, pp. 170-176, (2001).
[9] Y. Weissman, An improved analytical approximation to n!, Amer. J. Phys., 51, No. 9, (1983).
Published
2017-05-02
How to Cite
[1]
N. Batir, “Sharp inequalities for factorial n”, Proyecciones (Antofagasta, On line), vol. 27, no. 1, pp. 97-102, May 2017.
Issue
Section
Artículos
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
