FOUR LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS INVOLVING GAMMA FUNCTION |
Qi, Feng
(COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE HENAN UNIVERSITY, RESEARCH INSTITUTE OF MATHEMATICAL INEQUALITY THEORY HENAN POLYTECHNIC UNIVERSITY)
Niu, Da-Wei (COLLEGE OF INFORMATION AND BUSINESS ZHONGYUAN UNIVERSITY OF TECHNOLOGY) Cao, Jian (DEPARTMENT OF MATHEMATICS EAST CHINA NORMAL UNIVERSITY) Chen, Shou-Xin (COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE HEHAN UNIVERSITY) |
1 | H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181-221 DOI ScienceOn |
2 | 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 |
3 | 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 |
4 | 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 |
5 | 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 |
6 | 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 |
7 | Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v2n2/V2I2P8.tex |
8 | 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 |
9 | H. Alzer, Inequalities for the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 1, 141-147 DOI ScienceOn |
10 | H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389 DOI ScienceOn |
11 | H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337-346 DOI |
12 | H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460 |
13 | G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355-3362 DOI ScienceOn |
14 | A. Laforgia, Further inequalities for the gamma function, Math. Comp. 42 (1984), no. 166, 597-600 DOI |
15 | 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 |
16 | Available online at http://jipam.vu.edu.au/article.php?sid=574 |
17 | C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439 DOI |
18 | J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659-667 DOI |
19 | Ch.-P. Chen, Monotonicity and convexity for the gamma function, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 100 |
20 | 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 DOI ScienceOn |
21 | F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607 DOI ScienceOn |
22 | Available online at http://rgmia.vu.edu.au/v8n1.html |
23 | W. E. Clark and M. E. H. Ismail, Inequalities involving gamma and psi functions, Anal. Appl. (Singap.) 1 (2003), no. 1, 129-140 DOI |
24 | M. J. Cloud and B. C. Drachman, Inequalities with Applications to Engineering, Springer Verlag, 1998 |
25 | A. Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667-2673 DOI ScienceOn |
26 | N. Elezovic, C. Giordano, and J. Pecaric, The best bounds in Gautschi's inequality, Math. Inequal. Appl. 3 (2000), no. 2, 239-252 |
27 | A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), no. 1, 251-258 DOI |
28 | 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 DOI ScienceOn |
29 | 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 DOI ScienceOn |
30 | R. A. Horn, On infinitely divisible matrices, kernels, and functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 219-230 DOI |
31 | D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607-611 DOI |
32 | J. Lew, J. Frauenthal, and N. Keyfitz, On the average distances in a circular disc, SIAM Rev. 20 (1978), no. 3, 584-592 DOI ScienceOn |
33 | 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 |
34 | M. E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), no. 1-2, 54-63 DOI |
35 | 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 DOI ScienceOn |
36 | F. Qi, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions, Nonlinear Funct. Anal. Appl. 13 (2008), no. 1, in press |
37 | 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 DOI ScienceOn |
38 | F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), no. 7, 503-509 DOI ScienceOn |
39 | 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 |
40 | Available online at http://rgmia.vu.edu.au/v9n3.html |
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, J. Comput. Appl. Math. 212 (2008), no. 2, 444-456 DOI ScienceOn |
42 | 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 |
43 | Available online at http://rgmia.vu.edu.au/v10n2.html |
44 | 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 |
45 | Available online at http://rgmia.vu.edu.au/v7n1.html |
46 | 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 |
47 | 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 |
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 DOI |
51 | 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 DOI ScienceOn |
52 | J. Sandor, On certain inequalities for the Gamma function, RGMIA Res. Rep. Coll. 9 (2006), no. 1, Art. 11, 115-117 |
53 | Available online at http://rgmia.vu.edu.au/v9n1.html |
54 | 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 |
55 | 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) |
56 | J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), 563-564 DOI ScienceOn |
57 | D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941 |
58 | Available online at http://rgmia.vu.edu.au/v10n1.html |
59 | 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 |