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Some extensions of W. Gautschi's inequalities for the gamma function


Author: D. Kershaw
Journal: Math. Comp. 41 (1983), 607-611
MSC: Primary 33A15; Secondary 26D20, 65D20
DOI: https://doi.org/10.1090/S0025-5718-1983-0717706-5
MathSciNet review: 717706
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Abstract | References | Similar Articles | Additional Information

Abstract: It has been shown by W. Gautschi that if $ 0 < s < 1$, then for $ x \geqslant 1$

$\displaystyle {x^{1 - s}} < \frac{{\Gamma (x + 1)}}{{\Gamma (x + s)}} < \exp [(1 - s)\psi (x + 1)].$

The following closer bounds are proved:

$\displaystyle \exp [(1 - s)\psi (x + {s^{1/2}})] < \frac{{\Gamma (x + 1)}}{{\Ga... ...s)}} < \exp \left[ {(1 - s)\psi \left( {x + \frac{{s + 1}}{2}} \right)} \right]$

and

$\displaystyle {\left[ {x + \frac{s}{2}} \right]^{1 - s}} < \frac{{\Gamma (x + 1... ...x - \frac{1}{2} + {{\left( {s + \frac{1}{4}} \right)}^{1/2}}} \right]^{1 - s}}.$

These are compared with each other and with inequalities given by T. Erber and J. D. Kečkić and P. M. Vasić.


References [Enhancements On Off] (What's this?)

  • [1] E. F. Beckenbach & R. Bellman, Inequalities, 1st. ed., Springer-Verlag, Berlin and New York, 1961. MR 0158038 (28:1266)
  • [2] T. Erber, "The gamma function inequalities of Gurland and Gautschi," Skand. Aktuarietidskr., 1960, 1961, pp. 27-28. MR 0132846 (24:A2682)
  • [3] A. Erdélyi et al., Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
  • [4] W. Gautschi, "Some elementary inequalities relating to the gamma and incomplete gamma function," J. Math. Phys., v. 38, 1959, pp. 77-81. MR 0103289 (21:2067)
  • [5] J. D. Kečkić & P. M. Vasić, "Some inequalities for the gamma function," Publ. Inst. Math. (Beograd) (N.S.), v. 11, 1971, pp. 107-114. MR 0308446 (46:7560)
  • [6] D. S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin and New York, 1970. MR 0274686 (43:448)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1983-0717706-5
Keywords: Gamma function, inequalities
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society