메뉴 건너뛰기




Volumn 45, Issue 2, 2008, Pages 559-573

Four logarithmically completely monotonic functions involving gamma function

Author keywords

Completely monotonic function; Kershaw's inequality; Laforgia's inequality; Logarithmically completely monotonic function; Ratio of the gamma functions; Stirling's formula; Wendel's inequality

Indexed keywords


EID: 40949142017     PISSN: 03049914     EISSN: None     Source Type: Journal    
DOI: 10.4134/JKMS.2008.45.2.559     Document Type: Article
Times cited : (16)

References (50)
  • 1
    • 0003283055 scopus 로고
    • Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables
    • Superintendent of Documents, U.S. Government Printing Office, Washington, D.C
    • 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.
    • (1965) National Bureau of Standards Applied Mathematics Series , vol.55
    • Abramowitz, M.1    Stegun, I.A.2
  • 2
    • 22844454193 scopus 로고    scopus 로고
    • Inequalities for the gamma function
    • H. Alzer, Inequalities for the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 1, 141-147.
    • (2000) Proc. Amer. Math. Soc , vol.128 , Issue.1 , pp. 141-147
    • Alzer, H.1
  • 3
    • 0042376839 scopus 로고    scopus 로고
    • On some inequalities for the gamma and psi functions
    • _, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389.
    • (1997) Math. Comp , vol.66 , Issue.217 , pp. 373-389
    • Alzer, H.1
  • 4
    • 1042279113 scopus 로고    scopus 로고
    • Sharp inequalities for the digamma and polygamma functions
    • _, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181-221.
    • (2004) Forum Math , vol.16 , Issue.2 , pp. 181-221
    • Alzer, H.1
  • 5
    • 84968470274 scopus 로고
    • Some gamma function inequalities
    • _, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337-346.
    • (1993) Math. Comp , vol.60 , Issue.201 , pp. 337-346
    • Alzer, H.1
  • 6
    • 0036455848 scopus 로고    scopus 로고
    • Some classes of completely monotonic functions
    • H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460.
    • (2002) Ann. Acad. Sci. Fenn. Math , vol.27 , Issue.2 , pp. 445-460
    • Alzer, H.1    Berg, C.2
  • 7
    • 21944437101 scopus 로고    scopus 로고
    • A monotoneity property of the gamma function
    • G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355-3362.
    • (1997) Proc. Amer. Math. Soc , vol.125 , Issue.11 , pp. 3355-3362
    • Anderson, G.D.1    Qiu, S.-L.2
  • 8
    • 33646870267 scopus 로고
    • Some properties of a class of logarithmically completely monotonic functions
    • 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.
    • (1988) C. R. Acad. Bulgare Sci , vol.41 , Issue.2 , pp. 21-23
    • Atanassov, R.D.1    Tsoukrovski, U.V.2
  • 9
    • 33646380663 scopus 로고    scopus 로고
    • Integral representation of some functions related to the gamma function
    • C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439.
    • (2004) Mediterr. J. Math , vol.1 , Issue.4 , pp. 433-439
    • Berg, C.1
  • 10
    • 84966216618 scopus 로고
    • On gamma function inequalities
    • J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659-667.
    • (1986) Math. Comp , vol.47 , Issue.176 , pp. 659-667
    • Bustoz, J.1    Ismail, M.E.H.2
  • 11
    • 27844459163 scopus 로고    scopus 로고
    • Ch.-P. Chen, Monotonicity and convexity for the gamma function, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 100; Available online at http://jipam.vu.edu.au/ article.php?sid=574.
    • Ch.-P. Chen, Monotonicity and convexity for the gamma function, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 100; Available online at http://jipam.vu.edu.au/ article.php?sid=574.
  • 12
    • 33646338597 scopus 로고    scopus 로고
    • Logarithmically completely monotonic functions relating to the gamma function
    • 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.
    • (2006) J. Math. Anal. Appl , vol.321 , Issue.1 , pp. 405-411
    • Chen, C.-P.1    Qi, F.2
  • 13
    • 60149086015 scopus 로고    scopus 로고
    • Logarithmically complete monotonicity properties for the gamma functions
    • Art. 8; Available online at http://ajmaa. org/cgi-bin/paper.pl?string=v2n2/V2I2P8.tex
    • _, Logarithmically complete monotonicity properties for the gamma functions, Aust. J. Math. Anal. Appl. 2 (2005), no. 2, Art. 8; Available online at http://ajmaa. org/cgi-bin/paper.pl?string=v2n2/V2I2P8.tex.
    • (2005) Aust. J. Math. Anal. Appl , vol.2 , Issue.2
    • Chen, C.-P.1    Qi, F.2
  • 14
    • 35648954779 scopus 로고    scopus 로고
    • Logarithmically completely monotonic ratios of mean values and an application
    • _, Logarithmically completely monotonic ratios of mean values and an application, Glob. J. Math. Math. Sci. 1 (2005), no. 1, 71-76.
    • (2005) Glob. J. Math. Math. Sci , vol.1 , Issue.1 , pp. 71-76
    • Chen, C.-P.1    Qi, F.2
  • 15
    • 20144386265 scopus 로고    scopus 로고
    • Art. 18, 147-152; Available online at http://rgmia.vu.edu.au/v8nl.html
    • RGMIA Res. Rep. Coll. 8 (2005), no. 1, Art. 18, 147-152; Available online at http://rgmia.vu.edu.au/v8nl.html.
    • (2005) RGMIA Res. Rep. Coll , vol.8 , Issue.1
  • 16
    • 10344246460 scopus 로고    scopus 로고
    • Inequalities involving gamma and psi functions
    • W. E. Clark and M. E. H. Ismail, Inequalities involving gamma and psi functions, Anal. Appl. (Singap.) 1 (2003), no. 1, 129-140.
    • (2003) Anal. Appl. (Singap.) , vol.1 , Issue.1 , pp. 129-140
    • Clark, W.E.1    Ismail, M.E.H.2
  • 18
    • 23044521623 scopus 로고    scopus 로고
    • On some properties of the gamma function
    • Á. Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667-2673.
    • (2000) Proc. Amer. Math. Soc , vol.128 , Issue.9 , pp. 2667-2673
    • Elbert, A.1    Laforgia, A.2
  • 20
    • 0001308973 scopus 로고
    • Kolmogorov-Landau inequalities for monotone functions
    • A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), no. 1, 251-258.
    • (1982) J. Math. Anal. Appl , vol.90 , Issue.1 , pp. 251-258
    • Fink, A.M.1
  • 21
    • 33645771401 scopus 로고    scopus 로고
    • Completely monotonic functions involving the gamma and q-gamma functions
    • 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.
    • (2006) Proc. Amer. Math. Soc , vol.134 , Issue.4 , pp. 1153-1160
    • Grinshpan, A.Z.1    Ismail, M.E.H.2
  • 22
  • 23
    • 0022721836 scopus 로고
    • Completely monotonic functions associated with the gamma function and its q-analogues
    • 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.
    • (1986) J. Math. Anal. Appl , vol.116 , Issue.1 , pp. 1-9
    • Ismail, M.E.H.1    Lorch, L.2    Muldoon, M.E.3
  • 24
    • 84966228652 scopus 로고
    • Some extensions of W. Gautschi's inequalities for the gamma function
    • D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607-611.
    • (1983) Math. Comp , vol.41 , Issue.164 , pp. 607-611
    • Kershaw, D.1
  • 25
    • 84966209797 scopus 로고
    • Further inequalities for the gamma function
    • A. Laforgia, Further inequalities for the gamma function, Math. Comp. 42 (1984), no. 166, 597-600.
    • (1984) Math. Comp , vol.42 , Issue.166 , pp. 597-600
    • Laforgia, A.1
  • 26
    • 40949099089 scopus 로고
    • On the average distances in a circular disc
    • J. Lew, J. Frauenthal, and N. Keyfitz, On the average distances in a circular disc, SIAM Rev. 20 (1978), no. 3, 584-592.
    • (1978) SIAM Rev , vol.20 , Issue.3 , pp. 584-592
    • Lew, J.1    Frauenthal, J.2    Keyfitz, N.3
  • 27
    • 37249001973 scopus 로고    scopus 로고
    • Logarithmically complete monotonicity and Shur-convexity for some ratios of gamma functions
    • 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.
    • (2006) Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat , vol.17 , pp. 88-92
    • Li, A.-J.1    Zhao, W.-Z.2    Chen, C.-P.3
  • 28
    • 0003253630 scopus 로고
    • Formulas and Theorems for the Special Functions of Mathematical Physics
    • Third enlarged edition, Band Springer-Verlag New York, Inc, New York
    • 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.
    • (1966) Die Grundlehren der mathematischen Wissenschaften , vol.52
    • Magnus, W.1    Oberhettinger, F.2    Soni, R.P.3
  • 29
    • 0002146064 scopus 로고
    • Some monotonicity properties and characterizations of the gamma function
    • M. E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), no. 1-2, 54-63.
    • (1978) Aequationes Math , vol.18 , Issue.1-2 , pp. 54-63
    • Muldoon, M.E.1
  • 30
    • 34047143717 scopus 로고    scopus 로고
    • A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality
    • 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.
    • (2007) J. Comput. Appl. Math , vol.206 , Issue.2 , pp. 1007-1014
    • Qi, F.1
  • 31
    • 40949119865 scopus 로고    scopus 로고
    • Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions
    • in press
    • _, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions, Nonlinear Funct. Anal. Appl. 13 (2008), no. 1, in press.
    • (2008) Nonlinear Funct. Anal. Appl , vol.13 , Issue.1
    • Qi, F.1
  • 32
    • 34547224723 scopus 로고    scopus 로고
    • Three classes of logarithmically completely monotonic functions involving gamma and psi functions
    • _, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), no. 7, 503-509.
    • (2007) Integral Transforms Spec. Funct , vol.18 , Issue.7 , pp. 503-509
    • Qi, F.1
  • 33
    • 34249800432 scopus 로고    scopus 로고
    • Four logarithmically completely monotonic functions involving gamma function and originating from problems of traffic flow
    • Art. 9; Available online at http://rgmia.vu.edu.au/v9n3. html
    • 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; Available online at http://rgmia.vu.edu.au/v9n3. html.
    • (2006) RGMIA Res. Rep. Coll , vol.9 , Issue.3
    • Qi, F.1    Cao, J.2    Niu, D.-W.3
  • 34
    • 3442891734 scopus 로고    scopus 로고
    • A complete monotonicity property of the gamma function
    • F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607.
    • (2004) J. Math. Anal. Appl , vol.296 , Issue.2 , pp. 603-607
    • Qi, F.1    Chen, C.-P.2
  • 35
    • 34249093693 scopus 로고    scopus 로고
    • Logarithmically completely monotonic functions concerning gamma and digamma functions
    • 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.
    • (2007) Integral Transforms Spec. Funct , vol.18 , Issue.6 , pp. 435-443
    • Qi, F.1    Chen, S.-X.2    Cheung, W.-S.3
  • 36
    • 37249088817 scopus 로고    scopus 로고
    • A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality
    • 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.
    • (2008) J. Comput. Appl. Math , vol.212 , Issue.2 , pp. 444-456
    • Qi, F.1    Guo, B.-N.2
  • 37
    • 85036958579 scopus 로고    scopus 로고
    • Art. 5; Available online at http://rgmia.vu.edu.au/v10n2.html
    • RGMIA Res. Rep. Coll. 10 (2007), no. 2, Art. 5; Available online at http://rgmia.vu.edu.au/v10n2.html.
    • (2007) RGMIA Res. Rep. Coll , vol.10 , Issue.2
  • 38
    • 23844500554 scopus 로고    scopus 로고
    • Complete monotonicities of functions involving the gamma and digamma functions
    • Art. 8, 63-72; Available online at http://rgmia.vu.edu.au/v7n1.html
    • _, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63-72; Available online at http://rgmia.vu.edu.au/v7n1.html.
    • (2004) RGMIA Res. Rep. Coll , vol.7 , Issue.1
  • 39
    • 47849114054 scopus 로고    scopus 로고
    • Wendel-Gautschi-Kershaw's inequalities and sufficient and necessary conditions that a class of functions involving ratio of gamma functions are logarithmically completely monotonic
    • Art. 2; Available online at http://rgmia.vu.edu.au/v10n1.html
    • _, 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; Available online at http://rgmia.vu.edu.au/v10n1.html.
    • (2007) RGMIA Res. Rep. Coll , vol.10 , Issue.1
  • 40
    • 33747056370 scopus 로고    scopus 로고
    • The best bounds in Gautschi-Kershaw inequalities
    • 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.
    • (2006) Math. Inequal. Appl , vol.9 , Issue.3 , pp. 427-436
    • Qi, F.1    Guo, B.-N.2    Chen, C.-P.3
  • 41
    • 33645778397 scopus 로고    scopus 로고
    • Some completely monotonic functions involving the gamma and polygamma functions
    • Art. 5, 31-36; Available online at http://rgmia.vu.edu.au/v7n1.html
    • _, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 5, 31-36; Available online at http://rgmia.vu.edu.au/v7n1.html.
    • (2004) RGMIA Res. Rep. Coll , vol.7 , Issue.1
    • Qi, F.1    Guo, B.-N.2    Chen, C.-P.3
  • 42
    • 33645814909 scopus 로고    scopus 로고
    • Some completely monotonic functions involving the gamma and polygamma functions
    • _, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), no. 1, 81-88.
    • (2006) J. Aust. Math. Soc , vol.80 , Issue.1 , pp. 81-88
    • Qi, F.1    Guo, B.-N.2    Chen, C.-P.3
  • 43
    • 34249072000 scopus 로고    scopus 로고
    • Logarithmically completely monotonic functions involving gamma and polygamma functions
    • 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.
    • (2006) J. Math. Anal. Approx. Theory , vol.1 , Issue.1 , pp. 66-74
    • Qi, F.1    Niu, D.-W.2    Cao, J.3
  • 44
    • 33745628039 scopus 로고    scopus 로고
    • Two logarithmically completely monotonic functions connected with gamma function
    • 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.
    • (2006) Integral Transforms Spec. Funct , vol.17 , Issue.7 , pp. 539-542
    • Qi, F.1    Yang, Q.2    Li, W.3
  • 45
    • 78649791399 scopus 로고    scopus 로고
    • On certain inequalities for the Gamma function
    • Art. 11, 115-117; Available online at http://rgmia.vu.edu.au/v9n1 .html
    • J. Sándor, On certain inequalities for the Gamma function, RGMIA Res. Rep. Coll. 9 (2006), no. 1, Art. 11, 115-117; Available online at http://rgmia.vu.edu.au/v9n1 .html.
    • (2006) RGMIA Res. Rep. Coll , vol.9 , Issue.1
    • Sándor, J.1
  • 46
    • 0004149470 scopus 로고
    • Completely Monotonic and Related Functions
    • Report 93-108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands
    • 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.
    • (1993)
    • van Haeringen, H.1
  • 47
    • 85036912801 scopus 로고    scopus 로고
    • 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.
    • 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.
  • 49
    • 0041882790 scopus 로고
    • Note on the gamma function
    • J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), 563-564.
    • (1948) Amer. Math. Monthly , vol.55 , pp. 563-564
    • Wendel, J.G.1
  • 50
    • 0010119926 scopus 로고
    • The Laplace Transform
    • Princeton University Press, Princeton, N. J
    • D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941.
    • (1941) Princeton Mathematical Series , vol.6
    • Widder, D.V.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.