Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions

  • Authors

    • Bai-Ni Guo Henan Polytechnic University
    • Feng Qi Department of Mathematics, College of Science, Tianjin Polytechnic University,Tianjin City, 300160, China http://orcid.org/0000-0001-6239-2968
    2015-05-08
    https://doi.org/10.14419/gjma.v3i2.4605
  • Logarithmically complete monotonicity, Stieltjes function, Logarithmic function, Psi function, Power-exponential function, Conjecture

  • Abstract

    Let \(\Gamma\) and \(\psi=\frac{\Gamma'}{\Gamma}\) be respectively the classical Euler gamma function and the psi function and let \(\gamma=-\psi(1)=0.57721566\dotsc\) stand for the Euler-Mascheroni constant. In the paper, the authors simply confirm the logarithmically complete monotonicity of the power-exponential function \(q(t)=t^{t[\psi(t)-\ln t]-\gamma}\) on the unit interval \((0,1)\), concisely deny that \(q(t)\) is a Stieltjes function, surely point out fatal errors appeared in the paper [V. Krasniqi and A. Sh. Shabani, On a conjecture of a logarithmically completely monotonic function, Aust. J. Math. Anal. Appl. 11 (2014), no.1, Art.5, 5 pages; Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v11n1/V11I1P5.tex], and partially solve a conjecture posed in the article [B.-N. Guo, Y.-J. Zhang, and F. Qi, Refinements and sharpenings of some double inequalities for bounding the gamma function, J. Inequal. Pure Appl. Math. 9 (2008), no.1, Art.17; Available online at http://www.emis.de/journals/JIPAM/article953.html].

  • References

    1. [1] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373–389; Available online at http://dx.doi.org/10.1090/S0025-5718-97-00807-7.

      [2] 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.

      [3] C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433–439; Available online at http://dx.doi.org/10.1007/s00009-004-0022-6.

      [4] C. Berg and H. L. Pedersen, A completely monotonic function used in an inequality of Alzer, Comput. Methods Funct. Theory 12 (2012), no. 1, 329–341; Available online at http://dx.doi.org/10.1007/BF03321830.

      [5] C.-P. Chen and F. Qi, Completely monotonic function associated with the gamma function and proof of Wallis’ inequality, Tamkang J. Math. 36 (2005), no. 4, 303–307; Available online at http://dx.doi.org/10.5556/j.tkjm.36.2005.101.

      [6] B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21–30.

      [7] B.-N. Guo and F. Qi, Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/2.1.4607.8246.

      [8] B.-N. Guo and F. Qi, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Optim. Lett. 7 (2013), no. 6, 1139–1153; Available online at http://dx.doi.org/10.1007/s11590-012-0488-2.

      [9] B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1, 103–111; Available online at http://dx.doi.org/10.4134/bkms.2010.47.1.103.

      [10] B.-N. Guo, Y.-J. Zhang, and F. Qi, Refinements and sharpenings of some double inequalities for bounding the gamma function, J. Inequal. Pure Appl. Math. 9 (2008), no. 1, Art. 17; Available online at http://www.emis.de/journals/JIPAM/article953.html.

      [11] V. Krasniqi and A. Sh. Shabani, On a conjecture of a logarithmically completely monotonic function, Aust. J. Math. Anal. Appl. 11 (2014), no. 1, Art. 5, 5 pages; Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v11n1/V11I1P5.tex.

      [12] D. S. Mitrinović, J. E. PeÄarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.

      [13] F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603–607; Available online at http://dx.doi.org/10.1016/j.jmaa.2004.04.026.

      [14] 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.

      [15] F. Qi and B.-N. Guo, Monotonicity and logarithmic convexity relating to the volume of the unit ball, available online at http://arxiv.org/abs/0902.2509.

      [16] F. Qi, B.-N. Guo, and C.-P. Chen, 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.org/v7n1.php.

      [17] F. Qi, B.-N. Guo, and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), 81–88; Available online at http://dx.doi.org/10.1017/S1446788700011393.

      [18] F. Qi and W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, available online at http://arxiv.org/abs/1303.1877.

      [19] F. Qi, W. Li, and B.-N. Guo, Generalizations of a theorem of I. Schur, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 15; Available online at http://rgmia.org/v9n3.php.

      [20] 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.

      [21] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.

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  • How to Cite

    Guo, B.-N., & Qi, F. (2015). Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions. Global Journal of Mathematical Analysis, 3(2), 77-80. https://doi.org/10.14419/gjma.v3i2.4605

    Received date: 2015-04-09

    Accepted date: 2015-05-04

    Published date: 2015-05-08