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Mathematics of Computation

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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: 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:

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


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

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

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Keywords: Gamma function, inequalities
Article copyright: © Copyright 1983 American Mathematical Society

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