A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

  • Authors

    • Fang-Fang Liu Department of Mathematics, College of Science, Tianjin Polytechnic University,Tianjin City, 300160, China
    • Xiao-Ting Shi Department of Mathematics, College of Science, Tianjin Polytechnic University,Tianjin City, 300160, China
    • Feng Qi Department of Mathematics, College of Science, Tianjin Polytechnic University,Tianjin City, 300160, China http://orcid.org/0000-0001-6239-2968
    2015-09-08
    https://doi.org/10.14419/gjma.v3i4.5187
  • Necessary Condition, Sufficient Condition, Logarithmically Completely Monotonic Function, Gamma Function, Catalan Number, Catalan Function.

  • Abstract

    In the paper, the authors find necessary conditions and sufficient conditions for a function involving the gamma function and originating from investigation of properties of the Catalan numbers and function in combinatorics to be logarithmically completely monotonic.

  • References

    1. [1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974.

      [2] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics---A Foundation for Computer Science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994.

      [3] A.-Q. Liu, G.-F. Li, B.-N. Guo, and F. Qi, Monotonicity and logarithmic concavity of two functions involving exponential function, Internat. J. Math. Ed. Sci. Tech. 39 (2008), no. 5, 686--691; Available online at http://dx.doi.org/10.1080/00207390801986841.

      [4] D. S. Mitrinović, J. E. PeÄarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993; Available online at http://dx.doi.org/10.1007/978-94-017-1043-5.

      [5] F. Qi, A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers, ResearchGate Research, available online at http://dx.doi.org/10.13140/RG.2.1.1401.2009.

      [6] F. Qi, Asymptotic expansions, complete monotonicity, and inequalities of the Catalan numbers, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/RG.2.1.4371.6321.

      [7] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages; Available online at http://dx.doi.org/10.1155/2010/493058.

      [8] F. Qi, Bounds for the ratio of two gamma functions: from Gautschi's and Kershaw's inequalities to complete monotonicity, Turkish J. Anal. Number Theory 2 (2014), no. 5, 152--164; Available online at http://dx.doi.org/10.12691/tjant-2-5-1.

      [9] F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), no. 7, 503--509; Available online at http://dx.doi.org/10.1080/10652460701358976.

      [10] F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), 603--607; Available online at http://dx.doi.org/10.1016/j.jmaa.2004.04.026.

      [11] 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; Available online at http://rgmia.org/v7n1.php.

      [12] F. Qi, S. Guo, and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149--2160; Available online at http://dx.doi.org/10.1016/j.cam.2009.09.044.

      [13] F. Qi and W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, J. Appl. Anal. Comput. 5 (2015), no. 4, 626--634; Available online at http://dx.doi.org/10.11948/2015049.

      [14] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions---From Wendel's and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), no. 2, 132--158; Available online at http://dx.doi.org/10.15352/bjma/1342210165.

      [15] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel's asymptotic relation to Elezović-Giordano-PeÄarić's theorem, J. Inequal. Appl. 2013, 2013:542, 20 pages; Available online at http://dx.doi.org/10.1186/1029-242X-2013-542.

      [16] F. Qi, Q.-M. Luo, and B.-N. Guo, Complete monotonicity of a function involving the divided difference of digamma functions, Sci. China Math. 56 (2013), no. 11, 2315--2325; Available online at http://dx.doi.org/10.1007/s11425-012-4562-0.

      [17] F. Qi, Q.-M. Luo, and B.-N. Guo, The function : ratio's properties, In: Analytic Number Theory, Approximation Theory, and Special Functions, G. V. Milovanović and M. Th. Rassias (Eds), Springer, 2014, pp. 485--494; Available online at http://dx.doi.org/10.1007/978-1-4939-0258-3_16.

      [18] F. Qi, F.-F. Liu, and X.-T. Shi, Comments on two completely monotonic functions involving the -trigamma function, ResearchGate Dataset, available online at http://dx.doi.org/10.13140/RG.2.1.3267.4401.

      [19] F. Qi, M. Mahmoud, X.-T. Shi, and F.-F. Liu, Some properties of the Catalan-Qi function related to the Catalan numbers, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/RG.2.1.3810.7369.

      [20] F. Qi, X.-T. Shi, and F.-F. Liu, An exponential representation for a function involving the gamma function and originating from the Catalan numbers, ResearchGate Research, available online at http://dx.doi.org/10.13140/RG.2.1.1086.4486.

      [21] F. Qi, X.-T. Shi, and F.-F. Liu, An integral representation, complete monotonicity, and inequalities of the Catalan numbers, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/RG.2.1.3754.4806.

      [22] F. Qi, X.-T. Shi, and F.-F. Liu, Several formulas for special values of the Bell polynomials of the second kind and applications, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/RG.2.1.3230.1927.

      [23] F. Qi, X.-T. Shi, M. Mahmoud, and F.-F. Liu, Schur-convexity of the Catalan-Qi function, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/RG.2.1.2434.4802.

      [24] F. Qi, C.-F. Wei, and B.-N. Guo, Complete monotonicity of a function involving the ratio of gamma functions and applications, Banach J. Math. Anal. 6 (2012), no. 1, 35--44; Available online at http://dx.doi.org/10.15352/bjma/1337014663.

      [25] R. L. Schilling, R. Song, and Z. VondraÄek, Bernstein Functions---Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012; Available online at http://dx.doi.org/10.1515/9783110269338.

      [26] X.-T. Shi, F.-F. Liu, and F. Qi, An integral representation of the Catalan numbers, Glob. J. Math. Anal. 3 (2015), no. 3, 130--133; Available online at http://dx.doi.org/10.14419/gjma.v3i3.5055.

      [27] R. Stanley and E. W. Weisstein, Catalan Number, From MathWorld--A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/CatalanNumber.html.

      [28] D. V. Widder, The Laplace Transform, Princeton Mathematical Series 6, Princeton University Press, Princeton, N. J., 1941.

      [29] S.-Q. Zhang, B.-N. Guo, and F. Qi, A concise proof for properties of three functions involving the exponential function, Appl. Math. E-Notes 9 (2009), 177--183.

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  • How to Cite

    Liu, F.-F., Shi, X.-T., & Qi, F. (2015). A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function. Global Journal of Mathematical Analysis, 3(4), 140-144. https://doi.org/10.14419/gjma.v3i4.5187

    Received date: 2015-08-16

    Accepted date: 2015-09-07

    Published date: 2015-09-08