## Completely monotonic functions involving the gamma and $q$-gamma functions

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- by Arcadii Z. Grinshpan and Mourad E. H. Ismail PDF
- Proc. Amer. Math. Soc.
**134**(2006), 1153-1160 Request permission

## Abstract:

We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and $q$-gamma functions.## References

- Horst Alzer,
*On some inequalities for the gamma and psi functions*, Math. Comp.**66**(1997), no. 217, 373–389. MR**1388887**, DOI 10.1090/S0025-5718-97-00807-7 - Horst Alzer,
*Inequalities for the gamma function*, Proc. Amer. Math. Soc.**128**(2000), no. 1, 141–147. MR**1622757**, DOI 10.1090/S0002-9939-99-04993-X - Horst Alzer,
*On a gamma function inequality of Gautschi*, Proc. Edinb. Math. Soc. (2)**45**(2002), no. 3, 589–600. MR**1933741**, DOI 10.1017/S0013091501000943 - Horst Alzer,
*On Ramanujan’s double inequality for the gamma function*, Bull. London Math. Soc.**35**(2003), no. 5, 601–607. MR**1989488**, DOI 10.1112/S0024609303002261 - Horst Alzer,
*On Gautschi’s harmonic mean inequality for the gamma function*, J. Comput. Appl. Math.**157**(2003), no. 1, 243–249. MR**1996881**, DOI 10.1016/S0377-0427(03)00456-4 - H. Alzer and C. Berg, Some classes of completely monotonic functions, II,
*The Ramanujan Journal*, to appear. - George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958**, DOI 10.1017/CBO9781107325937 - C. Berg, Integral representation of some functions related to the gamma function,
*Mediterranean J. Math.*, to appear. - Joaquin Bustoz and Mourad E. H. Ismail,
*On gamma function inequalities*, Math. Comp.**47**(1986), no. 176, 659–667. MR**856710**, DOI 10.1090/S0025-5718-1986-0856710-6 - Jacques Dutka,
*On some gamma function inequalities*, SIAM J. Math. Anal.**16**(1985), no. 1, 180–185. MR**772877**, DOI 10.1137/0516013 - Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,
*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756** - William Feller,
*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154** - George Gasper and Mizan Rahman,
*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153** - 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 - Walter Gautschi,
*A harmonic mean inequality for the gamma function*, SIAM J. Math. Anal.**5**(1974), 278–281. MR**350077**, DOI 10.1137/0505030 - Arcadii Z. Grinshpan,
*Inequalities generated by the gamma function*, Adv. in Appl. Math.**29**(2002), no. 1, 115–125. MR**1921547**, DOI 10.1016/S0196-8858(02)00006-4 - Arcadii Z. Grinshpan,
*General inequalities, consequences and applications*, Adv. in Appl. Math.**34**(2005), no. 1, 71–100. MR**2102276**, DOI 10.1016/j.aam.2004.05.001 - A. Z. Grinshpan, An integral recursive inequality and applications,
*The Ramanujan Journal*, to appear. - G. H. Hardy, J. E. Littlewood, and G. Pólya,
*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395** - Roger A. Horn,
*On infinitely divisible matrices, kernels, and functions*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**8**(1967), 219–230. MR**217836**, DOI 10.1007/BF00531524 - 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**, DOI 10.1016/0022-247X(86)90042-9 - Mourad E. H. Ismail and Martin E. Muldoon,
*Inequalities and monotonicity properties for gamma and $q$-gamma functions*, Approximation and computation (West Lafayette, IN, 1993) Internat. Ser. Numer. Math., vol. 119, Birkhäuser Boston, Boston, MA, 1994, pp. 309–323. MR**1333625** - Donat K. Kazarinoff,
*On Wallis’ formula*, Edinburgh Math. Notes**1956**(1956), no. 40, 19–21. MR**82501** - Feng Qi, Bai-Ni Guo, and Chao-Ping Chen, Some completely monotonic functions involving the gamma and polygamma functions,
*RGMIA Res. Rep. Coll.***7**(1), Art. 5 (2004); available online at http://rgmia.vu.edu.au/v7n1.html. - Feng Qi and Bai-Ni Guo, Complete monotonicities of functions involving the gamma and digamma functions,
*RGMIA Res. Rep. Coll.***7**(1), Art. 8 (2004); available online at http://rgmia.vu.edu.au/v7n1.html. - G. N. Watson,
*A note on Gamma functions*, Proc. Edinburgh Math. Soc. (2)**11**(1958/59), no. Edinburgh Math. Notes 42 (misprinted 41) (1959), 7–9. MR**117358** - David Vernon Widder,
*The Laplace Transform*, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR**0005923**

## Additional Information

**Arcadii Z. Grinshpan**- Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
- Email: azg@math.usf.edu
**Mourad E. H. Ismail**- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- MR Author ID: 91855
- Email: ismail@math.ucf.edu
- Received by editor(s): April 12, 2004
- Received by editor(s) in revised form: October 28, 2004, and November 9, 2004
- Published electronically: September 28, 2005
- Additional Notes: The second author’s research was partially supported by NSF grant DMS 99-70865.
- Communicated by: Carmen C. Chicone
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 1153-1160 - MSC (2000): Primary 33B15; Secondary 26A48
- DOI: https://doi.org/10.1090/S0002-9939-05-08050-0
- MathSciNet review: 2196051