Browse > Article
http://dx.doi.org/10.4134/JKMS.2008.45.1.273

SOME COMPLETELY MONOTONIC FUNCTIONS INVOLVING THE GAMMA AND POLYGAMMA FUNCTIONS  

Li, Ai-Jun (Department of Mathematics Shanghai University)
Chen, Chao-Ping (School of Mathematics and Informatics Research Institute of Applied Mathematics Henan Polytechnic University)
Publication Information
Journal of the Korean Mathematical Society / v.45, no.1, 2008 , pp. 273-287 More about this Journal
Abstract
In this paper, some logarithmically completely monotonic, strongly completely monotonic and completely monotonic functions related to the gamma, digamma and polygamma functions are established. Several inequalities, whose bounds are best possible, are obtained.
Keywords
logarithmically completely monotonic; completely monotonic; strongly completely monotonic; Laplace transforms;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
Times Cited By Web Of Science : 7  (Related Records In Web of Science)
Times Cited By SCOPUS : 5
연도 인용수 순위
1 H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181-221   DOI   ScienceOn
2 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
3 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
4 H. Alzer, Inequalities for the volume of the unit ball in Rn, J. Math. Anal. Appl. 252 (2000), no. 1, 353-363   DOI   ScienceOn
5 H. Alzer, A power mean inequality for the gamma function, Monatsh. Math. 131 (2000), no. 3, 179-188   DOI
6 H. Alzer, Mean-value inequalities for the polygamma functions, Aequationes Math. 61 (2001), no. 1-2, 151-161   DOI
7 H. Alzer and C. Berg, Some classes of completely monotonic functions. II., Ramanujan J. 11 (2006), no. 2, 225-248   DOI
8 H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460
9 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, with corrections, Washington, 1965
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 and J. Wells, Inequalities for the polygamma functions, SIAM J. Math. Anal. 29 (1998), no. 6, 1459-1466   DOI   ScienceOn
12 G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1713-1723   DOI   ScienceOn
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 K. Ball, Completely monotonic rational functions and Hall's marriage theorem, J. Combin. Theory Ser. B 61 (1994), no. 1, 118-124   DOI   ScienceOn
15 C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439   DOI
16 C. Berg and H. L. Pedersen, A completely monotone function related to the gamma function, J. Comput. Appl. Math. 133 (2001), no. 1-2, 219-230   DOI   ScienceOn
17 J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659-667   DOI
18 C. Berg and H. L. Pedersen, Pick functions related to the gamma function, Rocky Mountain J. Math. 32 (2002), no. 2, 507-525   DOI   ScienceOn
19 K. H. Borgwardt, The simplex method, A probabilistic analysis. Algorithms and Combinatorics: Study and Research Texts, 1. Springer-Verlag, Berlin, 1987
20 A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203-1215   DOI
21 Chao-Ping Chen, Complete monotonicity properties for a ratio of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 16 (2005), 26-28
22 P. J. Davis, Leonhard Euler's integral: A historical profile of the gamma function, Amer. Math. Monthly 66 (1959), 849-869   DOI   ScienceOn
23 B. J. English and G. Rousseau, Bounds for certain harmonic sums, J. Math. Anal. Appl. 206 (1997), no. 2, 428-441   DOI   ScienceOn
24 R. A. Horn, On infinitely divisible matrices, kernels, and functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 219-230   DOI
25 D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607-611   DOI
26 C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math. 10 (1974), 152-164   DOI
27 M. Merkle, Gurland's ratio for the gamma function, Comput. Math. Appl. 49 (2005), no. 2-3, 389-406   DOI   ScienceOn
28 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
29 M. Merkle, On log-convexity of a ratio of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8 (1997), 114-119
30 M. Merkle, Convexity, Schur-convexity and bounds for the gamma function involving the digamma function, Rocky Mountain J. Math. 28 (1998), no. 3, 1053-1066   DOI
31 H. Minc and L. Sathre, Some inequalities involving $(r!)^{1/r}$, Proc. Edinburgh Math. Soc. (2) 14 (1964/1965), 41-46
32 F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), 63-72, Art. 8
33 S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions, with applications, Math. Comp. 74 (2005), no. 250, 723-742   DOI   ScienceOn
34 S. Y. Trimble, J. Wells, and F. T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal. 20 (1989), no. 5, 1255-1259   DOI
35 Zh.-X. Wang and D.-R. Guo, T`eshu Hanshu Gailun (Introduction to Special Function), The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000
36 D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941
37 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