DOI QR코드

DOI QR Code

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)
  • Published : 2008.03.31

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 $(-\frac{1}{2},\infty)$ or $(0,\infty)$; 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.

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
  2. H. Alzer, Inequalities for the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 1, 141-147 https://doi.org/10.1090/S0002-9939-99-04993-X
  3. H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389 https://doi.org/10.1090/S0025-5718-97-00807-7
  4. H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337-346 https://doi.org/10.2307/2153171
  5. H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460
  6. G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355-3362 https://doi.org/10.1090/S0002-9939-97-04152-X
  7. 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
  8. C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439 https://doi.org/10.1007/s00009-004-0022-6
  9. J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659-667 https://doi.org/10.2307/2008180
  10. Ch.-P. Chen, Monotonicity and convexity for the gamma function, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 100
  11. Available online at http://jipam.vu.edu.au/article.php?sid=574
  12. 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 https://doi.org/10.1016/j.jmaa.2005.08.056
  13. 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
  14. Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v2n2/V2I2P8.tex
  15. 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
  16. Available online at http://rgmia.vu.edu.au/v8n1.html
  17. W. E. Clark and M. E. H. Ismail, Inequalities involving gamma and psi functions, Anal. Appl. (Singap.) 1 (2003), no. 1, 129-140 https://doi.org/10.1142/S0219530503000041
  18. M. J. Cloud and B. C. Drachman, Inequalities with Applications to Engineering, Springer Verlag, 1998
  19. A. Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667-2673 https://doi.org/10.1090/S0002-9939-00-05520-9
  20. N. Elezovic, C. Giordano, and J. Pecaric, The best bounds in Gautschi's inequality, Math. Inequal. Appl. 3 (2000), no. 2, 239-252
  21. A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), no. 1, 251-258 https://doi.org/10.1016/0022-247X(82)90057-9
  22. 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 https://doi.org/10.1090/S0002-9939-05-08050-0
  23. R. A. Horn, On infinitely divisible matrices, kernels, and functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 219-230 https://doi.org/10.1007/BF00531524
  24. 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 https://doi.org/10.1016/0022-247X(86)90042-9
  25. D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607-611 https://doi.org/10.2307/2007697
  26. A. Laforgia, Further inequalities for the gamma function, Math. Comp. 42 (1984), no. 166, 597-600 https://doi.org/10.2307/2007604
  27. J. Lew, J. Frauenthal, and N. Keyfitz, On the average distances in a circular disc, SIAM Rev. 20 (1978), no. 3, 584-592 https://doi.org/10.1137/1020073
  28. 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
  29. M. E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), no. 1-2, 54-63 https://doi.org/10.1007/BF01844067
  30. 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 https://doi.org/10.1016/j.cam.2006.09.005
  31. F. Qi, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions, Nonlinear Funct. Anal. Appl. 13 (2008), no. 1, in press
  32. F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), no. 7, 503-509 https://doi.org/10.1080/10652460701358976
  33. 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
  34. Available online at http://rgmia.vu.edu.au/v9n3.html
  35. F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607 https://doi.org/10.1016/j.jmaa.2004.04.026
  36. 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 https://doi.org/10.1080/10652460701318418
  37. 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 https://doi.org/10.1016/j.cam.2006.12.022
  38. 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
  39. Available online at http://rgmia.vu.edu.au/v10n2.html
  40. 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
  41. Available online at http://rgmia.vu.edu.au/v7n1.html
  42. 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
  43. Available online at http://rgmia.vu.edu.au/v10n1.html
  44. 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
  45. 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
  46. Available online at http://rgmia.vu.edu.au/v7n1.html
  47. 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 https://doi.org/10.1017/S1446788700011393
  48. 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
  49. 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 https://doi.org/10.1080/10652460500422379
  50. J. Sandor, On certain inequalities for the Gamma function, RGMIA Res. Rep. Coll. 9 (2006), no. 1, Art. 11, 115-117
  51. Available online at http://rgmia.vu.edu.au/v9n1.html
  52. 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
  53. 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)
  54. J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), 563-564 https://doi.org/10.2307/2304460
  55. D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941
  56. H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181-221 https://doi.org/10.1515/form.2004.009
  57. 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
  58. 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
  59. 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

Cited by

  1. Logarithmically Complete Monotonicity Properties Relating to the Gamma Function vol.2011, 2011, https://doi.org/10.1155/2011/896483
  2. Asian options and meromorphic Lévy processes vol.18, pp.4, 2014, https://doi.org/10.1007/s00780-014-0237-8
  3. Bounds for the Ratio of Two Gamma Functions vol.2010, 2010, https://doi.org/10.1155/2010/493058
  4. Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks 2016, https://doi.org/10.1007/s11009-016-9503-x
  5. A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi–Kershaw’s inequality vol.224, pp.2, 2009, https://doi.org/10.1016/j.cam.2008.05.030
  6. Wendel’s and Gautschi’s inequalities: Refinements, extensions, and a class of logarithmically completely monotonic functions vol.205, pp.1, 2008, https://doi.org/10.1016/j.amc.2008.07.005
  7. A Class of Logarithmically Completely Monotonic Functions Associated with a Gamma Function vol.2010, pp.1, 2010, https://doi.org/10.1155/2010/392431