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Volumn 233, Issue 9, 2010, Pages 2149-2160

Complete monotonicity of some functions involving polygamma functions

Author keywords

Bernoulli numbers; Completely monotonic function; Infinite series; Monotonicity; Polygamma function

Indexed keywords

BERNOULLI NUMBERS; COMPLETELY MONOTONIC FUNCTION; INFINITE SERIES; MONOTONICITY; POLYGAMMA FUNCTION;

EID: 73249119072     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2009.09.044     Document Type: Article
Times cited : (65)

References (55)
  • 2
  • 3
    • 34250967495 scopus 로고
    • Summationsmethoden und Momentfolgen I
    • Hausdorff F. Summationsmethoden und Momentfolgen I. Math. Z. 9 (1921) 74-109
    • (1921) Math. Z. , vol.9 , pp. 74-109
    • Hausdorff, F.1
  • 4
    • 0003647259 scopus 로고
    • Springer, Berlin, Heidelberg, New York
    • Berg C., Christensen J.P.R., and Ressel P. Harmonic Analysis on Semigroups: Theory of Positive Deinite and Related Functions. Graduate Texts in Mathematics vol. 100 (1984), Springer, Berlin, Heidelberg, New York
    • (1984) Graduate Texts in Mathematics , vol.100
    • Berg, C.1    Christensen, J.P.R.2    Ressel, P.3
  • 5
    • 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
  • 6
    • 38149147370 scopus 로고
    • Completely monotonic rational functions and Hall's marriage theorem
    • Ball K. Completely monotonic rational functions and Hall's marriage theorem. J. Combin. Theory Ser. B 61 (1994) 118-124
    • (1994) J. Combin. Theory Ser. B , vol.61 , pp. 118-124
    • Ball, K.1
  • 7
    • 0001160530 scopus 로고
    • Error bounds for asymptotic expansions of the ratio of two gamma functions
    • Frenzen C.L. Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18 (1987) 890-896
    • (1987) SIAM J. Math. Anal. , vol.18 , pp. 890-896
    • Frenzen, C.L.1
  • 9
    • 0014926481 scopus 로고
    • On monotonicity of the relaxation functions of viscoelastic material
    • Day W.A. On monotonicity of the relaxation functions of viscoelastic material. Proc. Cambridge Philos. Soc. 67 (1970) 503-508
    • (1970) Proc. Cambridge Philos. Soc. , vol.67 , pp. 503-508
    • Day, W.A.1
  • 11
    • 0003352027 scopus 로고
    • Potential Theory on Locally Compact Abelian Groups
    • Springer, Berlin
    • Berg C., and Forst G. Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Math. vol. 87 (1975), Springer, Berlin
    • (1975) Ergebnisse der Math. , vol.87
    • Berg, C.1    Forst, G.2
  • 12
    • 0003536178 scopus 로고
    • Springer, New York
    • Bondesson L. Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics vol. 76 (1992), Springer, New York
    • (1992) Lecture Notes in Statistics , vol.76
    • Bondesson, L.1
  • 13
    • 0001502378 scopus 로고
    • A probabilistic interpretation of complete monotonicity
    • Kimberling C.H. A probabilistic interpretation of complete monotonicity. Aequationes Math. 10 (1974) 152-164
    • (1974) Aequationes Math. , vol.10 , pp. 152-164
    • Kimberling, C.H.1
  • 14
    • 0004245694 scopus 로고
    • Abramowitz M., and Stegun I.A. (Eds), A Wiley-Interscience Publication. Selected Government Publications. John Wiley & Sons, Inc., New York National Bureau of Standards, Washington, DC
    • In: Abramowitz M., and Stegun I.A. (Eds). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Reprint of the 1972 edition (1984), A Wiley-Interscience Publication. Selected Government Publications. John Wiley & Sons, Inc., New York National Bureau of Standards, Washington, DC
    • (1984) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Reprint of the 1972 edition
  • 15
    • 0004108644 scopus 로고
    • Translated from the Chinese by D.-R. Guo and X.-J. Xia, World Scientific Publishing, Singapore
    • Wang Zh.-X., and Guo D.-R. Special Functions. Translated from the Chinese by D.-R. Guo and X.-J. Xia (1989), World Scientific Publishing, Singapore
    • (1989) Special Functions
    • Wang, Zh.-X.1    Guo, D.-R.2
  • 17
    • 0001458111 scopus 로고    scopus 로고
    • Mean-value inequalities for the polygamma functions
    • Alzer H. Mean-value inequalities for the polygamma functions. Aequationes Math. 61 1 (2001) 151-161
    • (2001) Aequationes Math. , vol.61 , Issue.1 , pp. 151-161
    • Alzer, H.1
  • 18
    • 1042279113 scopus 로고    scopus 로고
    • Sharp inequalities for the digamma and polygamma functions
    • Alzer H. Sharp inequalities for the digamma and polygamma functions. Forum Math. 16 (2004) 181-221
    • (2004) Forum Math. , vol.16 , pp. 181-221
    • Alzer, H.1
  • 19
    • 73249141961 scopus 로고    scopus 로고
    • Two logarithmically completely monotonic functions connected with gamma function
    • Art. 13. Available online at http://www.staff.vu.edu.au/rgmia/v8n3.asp
    • Qi F., and Li W. Two logarithmically completely monotonic functions connected with gamma function. RGMIA Res. Rep. Coll. 8 3 (2005) 493-497. http://www.staff.vu.edu.au/rgmia/v8n3.asp Art. 13. Available online at http://www.staff.vu.edu.au/rgmia/v8n3.asp
    • (2005) RGMIA Res. Rep. Coll. , vol.8 , Issue.3 , pp. 493-497
    • Qi, F.1    Li, W.2
  • 20
    • 73249150789 scopus 로고    scopus 로고
    • A subadditive property of the digamma function
    • Art. 9. Available online at http://www.staff.vu.edu.au/rgmia/v8n3.asp
    • Gao P. A subadditive property of the digamma function. RGMIA Res. Rep. Coll. 8 3 (2005). http://www.staff.vu.edu.au/rgmia/v8n3.asp Art. 9. Available online at http://www.staff.vu.edu.au/rgmia/v8n3.asp
    • (2005) RGMIA Res. Rep. Coll. , vol.8 , Issue.3
    • Gao, P.1
  • 21
    • 73249117213 scopus 로고    scopus 로고
    • A class of completely monotonic functions involving the polygamma functions
    • Guo B.-N., Chen R.-J., and Qi F. A class of completely monotonic functions involving the polygamma functions. J. Math. Anal. Approx. Theory 1 2 (2006) 124-134
    • (2006) J. Math. Anal. Approx. Theory , vol.1 , Issue.2 , pp. 124-134
    • Guo, B.-N.1    Chen, R.-J.2    Qi, F.3
  • 22
    • 33745628039 scopus 로고    scopus 로고
    • Two logarithmically completely monotonic functions connected with gamma function
    • Qi F., Yang Q., and Li W. Two logarithmically completely monotonic functions connected with gamma function. Integral Transforms Spec. Funct. 17 7 (2006) 539-542
    • (2006) Integral Transforms Spec. Funct. , vol.17 , Issue.7 , pp. 539-542
    • Qi, F.1    Yang, Q.2    Li, W.3
  • 23
    • 35649001010 scopus 로고    scopus 로고
    • A function involving gamma function and having logarithmically absolute convexity
    • Qi F., Guo B.-N., Guo S., and Chen Sh.-X. A function involving gamma function and having logarithmically absolute convexity. Integral Transforms Spec. Funct. 18 11 (2007) 837-843
    • (2007) Integral Transforms Spec. Funct. , vol.18 , Issue.11 , pp. 837-843
    • Qi, F.1    Guo, B.-N.2    Guo, S.3    Chen, Sh.-X.4
  • 24
    • 47849114054 scopus 로고    scopus 로고
    • A class of k-log-convex functions and their applications to some special functions
    • Art. 21. Available online at http://www.staff.vu.edu.au/rgmia/v10n1.asp
    • Qi F., Guo S., and Guo B.-N. A class of k-log-convex functions and their applications to some special functions. RGMIA Res. Rep. Coll. 10 1 (2007). http://www.staff.vu.edu.au/rgmia/v10n1.asp Art. 21. Available online at http://www.staff.vu.edu.au/rgmia/v10n1.asp
    • (2007) RGMIA Res. Rep. Coll. , vol.10 , Issue.1
    • Qi, F.1    Guo, S.2    Guo, B.-N.3
  • 25
    • 40749136341 scopus 로고    scopus 로고
    • A class of k-log-convex functions and their applications to some special functions
    • Qi F., Guo S., Guo B.-N., and Chen Sh.-X. A class of k-log-convex functions and their applications to some special functions. Integral Transforms Spec. Funct. 19 3 (2008) 195-200
    • (2008) Integral Transforms Spec. Funct. , vol.19 , Issue.3 , pp. 195-200
    • Qi, F.1    Guo, S.2    Guo, B.-N.3    Chen, Sh.-X.4
  • 26
    • 47849114054 scopus 로고    scopus 로고
    • Note on a class of completely monotonic functions involving the polygamma functions
    • Art. 5. Available online at http://www.staff.vu.edu.au/rgmia/v10n1.asp
    • Qi F., Guo S., and Guo B.-N. Note on a class of completely monotonic functions involving the polygamma functions. RGMIA Res. Rep. Coll. 10 1 (2007). http://www.staff.vu.edu.au/rgmia/v10n1.asp Art. 5. Available online at http://www.staff.vu.edu.au/rgmia/v10n1.asp
    • (2007) RGMIA Res. Rep. Coll. , vol.10 , Issue.1
    • Qi, F.1    Guo, S.2    Guo, B.-N.3
  • 27
    • 47849122739 scopus 로고    scopus 로고
    • A new upper bound in the second Kershaw's double inequality and its generalizations
    • Available online at http://dx.doi.org/10.1016/j.cam.2007.07.037
    • Qi F., Guo S., and Chen Sh.-X. A new upper bound in the second Kershaw's double inequality and its generalizations. J. Comput. Appl. Math. 220 1-2 (2008) 111-118. http://dx.doi.org/10.1016/j.cam.2007.07.037 Available online at http://dx.doi.org/10.1016/j.cam.2007.07.037
    • (2008) J. Comput. Appl. Math. , vol.220 , Issue.1-2 , pp. 111-118
    • Qi, F.1    Guo, S.2    Chen, Sh.-X.3
  • 28
    • 48549090221 scopus 로고    scopus 로고
    • Refinements, extensions and generalizations of the second Kershaw's double inequality
    • Qi F., Li X.-A., and Chen Sh.-X. Refinements, extensions and generalizations of the second Kershaw's double inequality. Math. Inequal. Appl. 11 3 (2008) 457-465
    • (2008) Math. Inequal. Appl. , vol.11 , Issue.3 , pp. 457-465
    • Qi, F.1    Li, X.-A.2    Chen, Sh.-X.3
  • 29
    • 33645019224 scopus 로고    scopus 로고
    • Sub- and super-additive properties of Fejér's sine polynomial
    • Alzer H., and Koumandos S. Sub- and super-additive properties of Fejér's sine polynomial. Bull. London Math. Soc. 38 2 (2006) 261-268
    • (2006) Bull. London Math. Soc. , vol.38 , Issue.2 , pp. 261-268
    • Alzer, H.1    Koumandos, S.2
  • 30
    • 0141503441 scopus 로고    scopus 로고
    • A subadditive property of the gamma function
    • Alzer H., and Ruscheweyh S. A subadditive property of the gamma function. J. Math. Anal. Appl. 285 (2003) 564-577
    • (2003) J. Math. Anal. Appl. , vol.285 , pp. 564-577
    • Alzer, H.1    Ruscheweyh, S.2
  • 31
    • 0032654421 scopus 로고    scopus 로고
    • A submultiplicative property of the psi function
    • Alzer H., and Ruehr O.G. A submultiplicative property of the psi function. J. Comput. Appl. Math. 101 (1999) 53-60
    • (1999) J. Comput. Appl. Math. , vol.101 , pp. 53-60
    • Alzer, H.1    Ruehr, O.G.2
  • 32
    • 41849114947 scopus 로고    scopus 로고
    • Convexities of some functions involving the polygamma functions
    • Cao J., Niu D.-W., and Qi F. Convexities of some functions involving the polygamma functions. Appl. Math. E-Notes 8 (2008) 53-57
    • (2008) Appl. Math. E-Notes , vol.8 , pp. 53-57
    • Cao, J.1    Niu, D.-W.2    Qi, F.3
  • 33
    • 73249124628 scopus 로고    scopus 로고
    • F. Qi, B.-N. Guo, A note on additivity of polygamma functions. Available online at http://arxiv.org/abs/0903.0888
    • F. Qi, B.-N. Guo, A note on additivity of polygamma functions. Available online at http://arxiv.org/abs/0903.0888
  • 34
    • 47849114054 scopus 로고    scopus 로고
    • Subadditive and superadditive properties of polygamma functions
    • Art. 4. Available online at http://www.staff.vu.edu.au/rgmia/v10(E).asp
    • Qi F., and Guo B.-N. Subadditive and superadditive properties of polygamma functions. RGMIA Res. Rep. Coll. 10 Suppl. (2007). http://www.staff.vu.edu.au/rgmia/v10(E).asp Art. 4. Available online at http://www.staff.vu.edu.au/rgmia/v10(E).asp
    • (2007) RGMIA Res. Rep. Coll. , vol.10 , Issue.SUPPL
    • Qi, F.1    Guo, B.-N.2
  • 35
    • 73249136531 scopus 로고    scopus 로고
    • A double inequality for the gamma and psi functions
    • Zhang X.-M., and Chu Y.-M. A double inequality for the gamma and psi functions. Int. J. Mod. Math. 3 1 (2008) 67-73
    • (2008) Int. J. Mod. Math. , vol.3 , Issue.1 , pp. 67-73
    • Zhang, X.-M.1    Chu, Y.-M.2
  • 36
    • 34047117831 scopus 로고    scopus 로고
    • Geometric convexity of a function involving gamma function and applications to inequality theory
    • Art. 17. Available online at http://jipam.vu.edu.au/article.php?sid=830
    • Zhang X.-M., Xu T.-Q., and Situ L.-B. Geometric convexity of a function involving gamma function and applications to inequality theory. J. Inequal. Pure Appl. Math. 8 1 (2007). http://jipam.vu.edu.au/article.php%3Fsid%3D830 Art. 17. Available online at http://jipam.vu.edu.au/article.php?sid=830
    • (2007) J. Inequal. Pure Appl. Math. , vol.8 , Issue.1
    • Zhang, X.-M.1    Xu, T.-Q.2    Situ, L.-B.3
  • 37
    • 33646870267 scopus 로고
    • Some properties of a class of logarithmically completely monotonic functions
    • Atanassov R.D., and Tsoukrovski U.V. Some properties of a class of logarithmically completely monotonic functions. C.R. Acad. Bulgare Sci. 41 2 (1988) 21-23
    • (1988) C.R. Acad. Bulgare Sci. , vol.41 , Issue.2 , pp. 21-23
    • Atanassov, R.D.1    Tsoukrovski, U.V.2
  • 38
    • 33645766365 scopus 로고    scopus 로고
    • Complete monotonicities of functions involving the gamma and digamma functions
    • Art. 8. Available online at http://www.staff.vu.edu.au/rgmia/v7n1.asp
    • Qi F., and Guo B.-N. Complete monotonicities of functions involving the gamma and digamma functions. RGMIA Res. Rep. Coll. 7 1 (2004) 63-72. http://www.staff.vu.edu.au/rgmia/v7n1.asp Art. 8. Available online at http://www.staff.vu.edu.au/rgmia/v7n1.asp
    • (2004) RGMIA Res. Rep. Coll. , vol.7 , Issue.1 , pp. 63-72
    • Qi, F.1    Guo, B.-N.2
  • 39
    • 77749270240 scopus 로고    scopus 로고
    • F. Qi, B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (1) (2010) (in press)
    • F. Qi, B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (1) (2010) (in press)
  • 40
    • 33645778397 scopus 로고    scopus 로고
    • Some completely monotonic functions involving the gamma and polygamma functions
    • Art. 5. Available online at http://www.staff.vu.edu.au/rgmia/v7n1.asp
    • Qi F., Guo B.-N., and Chen Ch.-P. Some completely monotonic functions involving the gamma and polygamma functions. RGMIA Res. Rep. Coll. 7 1 (2004) 31-36. http://www.staff.vu.edu.au/rgmia/v7n1.asp Art. 5. Available online at http://www.staff.vu.edu.au/rgmia/v7n1.asp
    • (2004) RGMIA Res. Rep. Coll. , vol.7 , Issue.1 , pp. 31-36
    • Qi, F.1    Guo, B.-N.2    Chen, Ch.-P.3
  • 41
    • 33645814909 scopus 로고    scopus 로고
    • Some completely monotonic functions involving the gamma and polygamma functions
    • Qi F., Guo B.-N., and Chen Ch.-P. Some completely monotonic functions involving the gamma and polygamma functions. J. Aust. Math. Soc. 80 (2006) 81-88
    • (2006) J. Aust. Math. Soc. , vol.80 , pp. 81-88
    • Qi, F.1    Guo, B.-N.2    Chen, Ch.-P.3
  • 42
    • 3442891734 scopus 로고    scopus 로고
    • A complete monotonicity property of the gamma function
    • Qi F., and Chen Ch.-P. A complete monotonicity property of the gamma function. J. Math. Anal. Appl. 296 (2004) 603-607
    • (2004) J. Math. Anal. Appl. , vol.296 , pp. 603-607
    • Qi, F.1    Chen, Ch.-P.2
  • 43
    • 33646380663 scopus 로고    scopus 로고
    • Integral representation of some functions related to the gamma function
    • Berg C. Integral representation of some functions related to the gamma function. Mediterr. J. Math. 1 4 (2004) 433-439
    • (2004) Mediterr. J. Math. , vol.1 , Issue.4 , pp. 433-439
    • Berg, C.1
  • 45
    • 33645771401 scopus 로고    scopus 로고
    • Completely monotonic functions involving the gamma and q-gamma functions
    • Grinshpan A.Z., and Ismail M.E.H. Completely monotonic functions involving the gamma and q-gamma functions. Proc. Amer. Math. Soc.. 134 (2006) 1153-1160
    • (2006) Proc. Amer. Math. Soc.. , vol.134 , pp. 1153-1160
    • Grinshpan, A.Z.1    Ismail, M.E.H.2
  • 46
    • 37249046597 scopus 로고    scopus 로고
    • Generalizations of a theorem of I. Schur
    • Art. 15. Available online at http://www.staff.vu.edu.au/rgmia/v9n3.asp
    • Qi F., Li W., and Guo B.-N. Generalizations of a theorem of I. Schur. RGMIA Res. Rep. Coll. 9 3 (2006). http://www.staff.vu.edu.au/rgmia/v9n3.asp Art. 15. Available online at http://www.staff.vu.edu.au/rgmia/v9n3.asp
    • (2006) RGMIA Res. Rep. Coll. , vol.9 , Issue.3
    • Qi, F.1    Li, W.2    Guo, B.-N.3
  • 47
    • 33645140646 scopus 로고
    • On infinitely divisible matrices, kernels and functions
    • Horn R.A. On infinitely divisible matrices, kernels and functions. Z. Wahrscheinlichkeitstheor. Verwandte. Geb 8 (1967) 219-230
    • (1967) Z. Wahrscheinlichkeitstheor. Verwandte. Geb , vol.8 , pp. 219-230
    • Horn, R.A.1
  • 49
    • 0002146064 scopus 로고
    • Some monotonicity properties and characterizations of the gamma function
    • Muldoon M.E. Some monotonicity properties and characterizations of the gamma function. Aequationes Math. 18 (1978) 54-63
    • (1978) Aequationes Math. , vol.18 , pp. 54-63
    • Muldoon, M.E.1
  • 50
    • 36348957497 scopus 로고    scopus 로고
    • A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums
    • Qi F. A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums. ANZIAM J. 48 4 (2007) 523-532
    • (2007) ANZIAM J. , vol.48 , Issue.4 , pp. 523-532
    • Qi, F.1
  • 51
    • 34047144395 scopus 로고    scopus 로고
    • A completely monotonic function involving divided differences of psi and polygamma functions and an application
    • Art. 8. Available online at http://www.staff.vu.edu.au/rgmia/v9n4.asp
    • Qi F. A completely monotonic function involving divided differences of psi and polygamma functions and an application. RGMIA Res. Rep. Coll. 9 4 (2006). http://www.staff.vu.edu.au/rgmia/v9n4.asp Art. 8. Available online at http://www.staff.vu.edu.au/rgmia/v9n4.asp
    • (2006) RGMIA Res. Rep. Coll. , vol.9 , Issue.4
    • Qi, F.1
  • 52
    • 34047144395 scopus 로고    scopus 로고
    • The best bounds in Kershaw's inequality and two completely monotonic functions
    • Art. 2. Available online at http://www.staff.vu.edu.au/rgmia/v9n4.asp
    • Qi F. The best bounds in Kershaw's inequality and two completely monotonic functions. RGMIA Res. Rep. Coll. 9 4 (2006). http://www.staff.vu.edu.au/rgmia/v9n4.asp Art. 2. Available online at http://www.staff.vu.edu.au/rgmia/v9n4.asp
    • (2006) RGMIA Res. Rep. Coll. , vol.9 , Issue.4
    • Qi, F.1
  • 53
    • 34547224723 scopus 로고    scopus 로고
    • Three classes of logarithmically completely monotonic functions involving gamma and psi functions
    • Qi F. Three classes of logarithmically completely monotonic functions involving gamma and psi functions. Integral Transforms Spec. Funct. 18 7 (2007) 503-509
    • (2007) Integral Transforms Spec. Funct. , vol.18 , Issue.7 , pp. 503-509
    • Qi, F.1
  • 54
    • 73249123569 scopus 로고    scopus 로고
    • Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications
    • Available online at http://dx.doi.org/10.3934/cpaa.2009.8.1975
    • Qi F., and Guo B.-N. Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Commun. Pure Appl. Anal. 8 6 (2009) 1975-1989. http://dx.doi.org/10.3934/cpaa.2009.8.1975 Available online at http://dx.doi.org/10.3934/cpaa.2009.8.1975
    • (2009) Commun. Pure Appl. Anal. , vol.8 , Issue.6 , pp. 1975-1989
    • Qi, F.1    Guo, B.-N.2
  • 55
    • 33646338597 scopus 로고    scopus 로고
    • Logarithmically completely monotonic functions relating to the gamma function
    • Chen Ch.-P., and Qi F. Logarithmically completely monotonic functions relating to the gamma function. J. Math. Anal. Appl. 321 1 (2006) 405-411
    • (2006) J. Math. Anal. Appl. , vol.321 , Issue.1 , pp. 405-411
    • Chen, Ch.-P.1    Qi, F.2


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