On gamma function inequalities

Authors:
Joaquin Bustoz and Mourad E. H. Ismail

Journal:
Math. Comp. **47** (1986), 659-667

MSC:
Primary 33A15; Secondary 26D20

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856710-6

MathSciNet review:
856710

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that certain functions involving quotients of gamma functions are completely monotonic. This leads to inequalities involving gamma functions. We also establish the infinite divisibility of several probability distributions whose Laplace transforms involve quotients of gamma functions.

**[1]**J. Dutka, "On some gamma function inequalities,"*SIAM J. Math. Anal.*, v. 16, 1985, pp. 180-185. MR**772877 (86e:33002)****[2]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Higher Transcendental Functions*, Vol. 1, McGraw-Hill, New York, 1953.**[3]**W. Feller,*An Introduction to Probability Theory and its Applications*, Wiley, New York, 1966. MR**0210154 (35:1048)****[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]**M. E. H. Ismail, L. Lorch & M. Muldoon, "Completely monotonic functions associated with the gamma function and its*q*-analogues,"*J. Math. Anal. Appl.*, v. 116, 1986, pp. 1-9. MR**837337 (88b:33002)****[6]**D. K. Kazarinoff, "On Wallis' formula,"*Edinburgh. Math. Soc. Notes*, No. 40, 1956, pp. 19-21. MR**0082501 (18:560g)****[7]**A. Kemp, "On gamma function inequalities,"*Skand. Aktuarietidskr.*, v. 56, 1973, pp. 65-69. MR**0345271 (49:10007)****[8]**D. Kershaw, "Some extensions of W. Gautschi's inequalities for the gamma function,"*Math. Comp.*, v. 41, 1983, pp 607-611. MR**717706 (84m:33003)****[9]**G. N. Watson, "A note on gamma functions,"*Proc. Edinburgh Math. Soc.*(2), v. 11, 1958/1959,*Edinburgh Math. Soc. Notes*, No. 42, 1959, pp. 7-9. MR**0117358 (22:8138)****[10]**D. V. Widder,*The Laplace Transform*, Princeton Univ. Press, Princeton, N.J., 1944.**[11]**J. Wimp,*Sequence Transformations and Their Applications*, Academic Press, New York, 1981. MR**615250 (84e:65005)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856710-6

Keywords:
Gamma function,
psi function,
complete monotonicity,
inequalities,
infinite divisibility,
Laplace transform

Article copyright:
© Copyright 1986
American Mathematical Society