Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions
About this article
Keywords:
Logarithmically complete monotonicity, Stieltjes function, Logarithmic function, Psi function, Power-exponential function, ConjectureAbstract
Let Γ denote the classical Euler gamma function, let ψ = Γ′/Γ denote the psi function, and let γ = −ψ(1) = 0.57721566... be the Euler-Mascheroni constant. This paper studies the logarithmically complete monotonicity of the power-exponential function q(t) = t^{t[ψ(t) − ln t] − γ} on the interval (0, 1). It proves that q(t) is logarithmically completely monotonic on this interval and shows that q(t) is not a Stieltjes function. The paper also revisits related assertions in previous work and provides corrections and refinements to results concerning logarithmically completely monotonic functions and inequalities associated with the gamma function.
References
[1] 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
[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. https://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. https://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. https://doi.org/10.5556/j.tkjm.36.2005.101
View more references (16)
[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. https://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. https://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. https://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. 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. http://ajmaa.org/cgi-bin/paper.pl?string=v11n1/V11I1P5.tex
[12] D. S. Mitrinović, J. E. Pečarić, 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. https://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. http://rgmia.org/v7n1.php
[15] F. Qi and B.-N. Guo, Monotonicity and logarithmic convexity relating to the volume of the unit ball. 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. 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. https://doi.org/10.1017/S1446788700011393
[18] F. Qi and W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions. 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. http://rgmia.org/v9n3.php
[20] R. L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions: Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012. https://doi.org/10.1515/9783110269338
[21] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.