Complete monotonicity of functions connected with the exponential function and derivatives. (English) Zbl 1474.33005

Summary: Some complete monotonicity results that the functions \(\pm 1 / \left(e^{\pm t} - 1\right)\) are logarithmically completely monotonic, and that differences between consecutive derivatives of these two functions are completely monotonic, and that the ratios between consecutive derivatives of these two functions are decreasing on \(\left(0, \infty\right)\) are discovered. As applications of these newly discovered results, some complete monotonicity results concerning the polylogarithm are found. Finally a conjecture on the complete monotonicity of the above-mentioned ratios is posed.


33B10 Exponential and trigonometric functions
26A48 Monotonic functions, generalizations
33B30 Higher logarithm functions
Full Text: DOI


[1] Guo, B.-N.; Qi, F., Some identities and an explicit formula for Bernoulli and Stirling numbers, Journal of Computational and Applied Mathematics, 255, 568-579 (2014) · Zbl 1291.11051
[2] Qi, F., Explicit formulas for computing Euler polynomials in terms of the second kind Stirling numbers
[3] Xu, A.-M.; Cen, Z.-D., Some identities involving exponential functions and Stirling numbers and applications, Journal of Computational and Applied Mathematics, 260, 201-207 (2014) · Zbl 1293.05011
[4] Mitrinović, D. S.; Pečarić, J. E.; Fink, A. M., Classical and New Inequalities in Analysis, 61 (1993), Kluwer Academic Publishers · Zbl 0771.26009
[5] Widder, D. V., The Laplace Transform (1946), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · JFM 67.0384.01
[6] Schilling, R. L.; Song, R.; Vondraček, Z., Bernstein Functions. Bernstein Functions, de Gruyter Studies in Mathematics, 37 (2010), Berlin, Germany: De Gruyter, Berlin, Germany
[7] Atanassov, R. D.; Tsoukrovski, U. V., Some properties of a class of logarithmically completely monotonic functions, Comptes Rendus de l’Académie Bulgare des Sciences, 41, 2, 21-23 (1988) · Zbl 0658.26010
[8] Berg, C., Integral representation of some functions related to the gamma function, Mediterranean Journal of Mathematics, 1, 4, 433-439 (2004) · Zbl 1162.33300
[9] Qi, F.; Luo, Q. M., Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezovic-Giordano-Pecaric’s theorem, Journal of Inequalities and Applications, 2013, article 542 (2013) · Zbl 1294.33004
[10] Qi, F.; Luo, Q.-M., Bounds for the ratio of two gamma functions: from Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach Journal of Mathematical Analysis, 6, 2, 132-158 (2012) · Zbl 1245.33004
[11] Koumandos, S.; Pedersen, H. L., On the asymptotic expansion of the logarithm of Barnes triple gamma function, Mathematica Scandinavica, 105, 2, 287-306 (2009) · Zbl 1184.33002
[12] Wikipedia, The Free Encyclopedia
[13] Qi, F.; Guo, B.-N., Some logarithmically completely monotonic functions related to the gamma function, Journal of the Korean Mathematical Society, 47, 6, 1283-1297 (2010) · Zbl 1208.26017
[14] Bochner, S., Harmonic Analysis and the Theory of Probability. Harmonic Analysis and the Theory of Probability, California Monographs in Mathematical Sciences (1955), Berkeley, Calif, USA: University of California Press, Berkeley, Calif, USA · Zbl 0068.11702
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