Sharp inequalities for factorial n

Authors

  • Necdet Batir Yunzuncu Yil University.

DOI:

https://doi.org/10.4067/S0716-09172008000100006

Keywords:

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.

Author Biography

Necdet Batir, Yunzuncu Yil University.

Faculty of Arts and Sciences.
Department of Mathematics.

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).

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