Some extensions of W. Gautschi’s inequalities for the gamma function
HTML articles powered by AMS MathViewer
- by D. Kershaw PDF
- Math. Comp. 41 (1983), 607-611 Request permission
Abstract:
It has been shown by W. Gautschi that if $0 < s < 1$, then for $x \geqslant 1$ \[ {x^{1 - s}} < \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} < \exp [(1 - s)\psi (x + 1)].\] The following closer bounds are proved: \[ \exp [(1 - s)\psi (x + {s^{1/2}})] < \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} < \exp \left [ {(1 - s)\psi \left ( {x + \frac {{s + 1}}{2}} \right )} \right ]\] and \[ {\left [ {x + \frac {s}{2}} \right ]^{1 - s}} < \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} < {\left [ {x - \frac {1}{2} + {{\left ( {s + \frac {1}{4}} \right )}^{1/2}}} \right ]^{1 - s}}.\] These are compared with each other and with inequalities given by T. Erber and J. D. Kečkić and P. M. Vasić.References
- Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 30, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0158038, DOI 10.1007/978-3-642-64971-4
- Thomas Erber, The gamma function inequalities of Gurland and Gautschi, Skand. Aktuarietidskr. 1960 (1960), 27–28 (1961). MR 132846, DOI 10.1080/03461238.1960.10410596 A. Erdélyi et al., Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
- Walter Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. and Phys. 38 (1959/60), 77–81. MR 103289, DOI 10.1002/sapm195938177
- 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 308446
- D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686, DOI 10.1007/978-3-642-99970-3
Additional Information
- © Copyright 1983 American Mathematical Society
- 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