FOUR LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS INVOLVING GAMMA FUNCTION

- Journal title : Journal of the Korean Mathematical Society
- Volume 45, Issue 2, 2008, pp.559-573
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2008.45.2.559

Title & Authors

FOUR LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS INVOLVING GAMMA FUNCTION

Qi, Feng; Niu, Da-Wei; Cao, Jian; Chen, Shou-Xin;

Qi, Feng; Niu, Da-Wei; Cao, Jian; Chen, Shou-Xin;

Abstract

In this paper, two classes of functions, involving a parameter and the classical Euler gamma function, and two functions, involving the classical Euler gamma function, are verified to be logarithmically completely monotonic in or ; some inequalities involving the classical Euler gamma function are deduced and compared with those originating from certain problems of traffic flow, due to J. Wendel and A. Laforgia, and relating to the well known Stirling's formula.

Keywords

logarithmically completely montonic function;completely monotonic function;ratio of the gamma functions;Kershaw's inequality;Laforgia's inequality;Stirling's formula;Wendel's inequality;

Language

English

Cited by

1.

2.

3.

4.

5.

References

1.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1965

3.

H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389

4.

H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181-221

6.

H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460

7.

G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355-3362

8.

R. D. Atanassov and U. V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988), no. 2, 21-23

9.

C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439

10.

J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659-667

11.

Ch.-P. Chen, Monotonicity and convexity for the gamma function, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 100

12.

Available online at http://jipam.vu.edu.au/article.php?sid=574

13.

Ch.-P. Chen and F. Qi, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (2006), no. 1, 405-411

14.

Ch.-P. Chen and F. Qi, Logarithmically complete monotonicity properties for the gamma functions, Aust. J. Math. Anal. Appl. 2 (2005), no. 2, Art. 8

15.

Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v2n2/V2I2P8.tex

16.

Ch.-P. Chen and F. Qi, Logarithmically completely monotonic ratios of mean values and an application, Glob. J. Math. Math. Sci. 1 (2005), no. 1, 71-76

17.

Ch.-P. Chen and F. Qi, Logarithmically completely monotonic ratios of mean values and an application, RGMIA Res. Rep. Coll. 8 (2005), no. 1, Art. 18, 147-152

18.

Available online at http://rgmia.vu.edu.au/v8n1.html

19.

W. E. Clark and M. E. H. Ismail, Inequalities involving gamma and psi functions, Anal. Appl. (Singap.) 1 (2003), no. 1, 129-140

20.

M. J. Cloud and B. C. Drachman, Inequalities with Applications to Engineering, Springer Verlag, 1998

21.

A. Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667-2673

22.

N. Elezovic, C. Giordano, and J. Pecaric, The best bounds in Gautschi's inequality, Math. Inequal. Appl. 3 (2000), no. 2, 239-252

23.

A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), no. 1, 251-258

24.

A. Z. Grinshpan and M. E. H. Ismail, Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1153-1160

25.

R. A. Horn, On infinitely divisible matrices, kernels, and functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 219-230

26.

M. E. H. Ismail, L. Lorch, and M. E. Muldoon, Completely monotonic functions associated with the gamma function and its q-analogues, J. Math. Anal. Appl. 116 (1986), no. 1, 1-9

27.

D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607-611

28.

29.

J. Lew, J. Frauenthal, and N. Keyfitz, On the average distances in a circular disc, SIAM Rev. 20 (1978), no. 3, 584-592

30.

A.-J. Li, W.-Zh. Zhao, and Ch.-P. Chen, Logarithmically complete monotonicity and Shur-convexity for some ratios of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 88-92

31.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York 1966

32.

M. E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), no. 1-2, 54-63

33.

F. Qi, A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality, J. Comput. Appl. Math. 206 (2007), no. 2, 1007-1014

34.

F. Qi, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions, Nonlinear Funct. Anal. Appl. 13 (2008), no. 1, in press

35.

F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), no. 7, 503-509

36.

F. Qi, J. Cao, and D.-W. Niu, Four logarithmically completely monotonic functions involving gamma function and originating from problems of traffic flow, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 9

37.

Available online at http://rgmia.vu.edu.au/v9n3.html

38.

F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607

39.

F. Qi, Sh.-X. Chen, and W.-S. Cheung, Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transforms Spec. Funct. 18 (2007), no. 6, 435-443

40.

F. Qi and B.-N. Guo, A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality, J. Comput. Appl. Math. 212 (2008), no. 2, 444-456

41.

F. Qi and B.-N. Guo, A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality, RGMIA Res. Rep. Coll. 10 (2007), no. 2, Art. 5

42.

Available online at http://rgmia.vu.edu.au/v10n2.html

43.

F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63-72

44.

Available online at http://rgmia.vu.edu.au/v7n1.html

45.

F. Qi and B.-N. Guo, Wendel-Gautschi-Kershaw's inequalities and sufficient and necessary conditions that a class of functions involving ratio of gamma functions are logarithmically completely monotonic, RGMIA Res. Rep. Coll. 10 (2007), no. 1, Art. 2

46.

Available online at http://rgmia.vu.edu.au/v10n1.html

47.

F. Qi, B.-N. Guo, and Ch.-P. Chen, The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl. 9 (2006), no. 3, 427-436

48.

F. Qi, B.-N. Guo, and Ch.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 5, 31-36

49.

Available online at http://rgmia.vu.edu.au/v7n1.html

50.

F. Qi, B.-N. Guo, and Ch.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), no. 1, 81-88

51.

F. Qi, D.-W. Niu, and J. Cao, Logarithmically completely monotonic functions involving gamma and polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), no. 1, 66-74

52.

F. Qi, Q. Yang, and W. Li, Two logarithmically completely monotonic functions connected with gamma function, Integral Transforms Spec. Funct. 17 (2006), no. 7, 539-542

53.

J. Sandor, On certain inequalities for the Gamma function, RGMIA Res. Rep. Coll. 9 (2006), no. 1, Art. 11, 115-117

54.

Available online at http://rgmia.vu.edu.au/v9n1.html

55.

H. van Haeringen, Completely Monotonic and Related Functions, Report 93-108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1993

56.

Zh.-X. Wang and D.-R. Guo, Special Functions, Translated from the Chinese by Guo and X. J. Xia. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989

57.

Zh.-X. Wang and D.-R. Guo, Teshu Hanshu Gailun, The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000. (Chinese)

59.

D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941