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

Full-text PDF

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]**Jacques Dutka,*On some gamma function inequalities*, SIAM J. Math. Anal.**16**(1985), no. 1, 180–185. MR**772877**, https://doi.org/10.1137/0516013**[2]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Higher Transcendental Functions*, Vol. 1, McGraw-Hill, New York, 1953.**[3]**William Feller,*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154****[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**, https://doi.org/10.1002/sapm195938177**[5]**Mourad E. H. Ismail, Lee Lorch, and Martin E. Muldoon,*Completely monotonic functions associated with the gamma function and its 𝑞-analogues*, J. Math. Anal. Appl.**116**(1986), no. 1, 1–9. MR**837337**, https://doi.org/10.1016/0022-247X(86)90042-9**[6]**Donat K. Kazarinoff,*On Wallis’ formula*, Edinburgh Math. Notes**1956**(1956), no. 40, 19–21. MR**0082501****[7]**Adrienne W. Kemp,*On gamma function inequalities*, Skand. Aktuarietidskr. (1973), 65–69. MR**0345271****[8]**D. Kershaw,*Some extensions of W. Gautschi’s inequalities for the gamma function*, Math. Comp.**41**(1983), no. 164, 607–611. MR**717706**, https://doi.org/10.1090/S0025-5718-1983-0717706-5**[9]**G. N. Watson,*A note on Gamma functions*, Proc. Edinburgh Math. Soc. (2)**11**(1958/1959), no. Edinburgh Math. Notes 42 (misprinted 41) (1959), 7–9. MR**0117358****[10]**D. V. Widder,*The Laplace Transform*, Princeton Univ. Press, Princeton, N.J., 1944.**[11]**Jet Wimp,*Sequence transformations and their applications*, Mathematics in Science and Engineering, vol. 154, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**615250**

Retrieve articles in *Mathematics of Computation*
with MSC:
33A15,
26D20

Retrieve articles in all journals with MSC: 33A15, 26D20

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