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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
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] Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 30, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0158038
  • [2] Thomas Erber, The gamma function inequalities of Gurland and Gautschi, Skand. Aktuarietidskr. 1960 (1960), 27–28 (1961). MR 0132846
  • [3] A. Erdélyi et al., Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
  • [4] Walter Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. and Phys. 38 (1959/60), 77–81. MR 0103289
  • [5] Jovan D. Kečkić and Petar M. Vasić, Some inequalities for the gamma function, Publ. Inst. Math. (Beograd) (N.S.) 11(25) (1971), 107–114. MR 0308446
  • [6] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686

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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