Completely monotonic functions involving the gamma and 𝑞-gamma functions

Grinshpan, Arcadii; Ismail, Mourad

doi:10.1090/S0002-9939-05-08050-0

Completely monotonic functions involving the gamma and $q$-gamma functions
HTML articles powered by AMS MathViewer

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