Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions
DOI:
https://doi.org/10.14419/gjma.v2i3.2919Abstract
In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-\psi'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind. These results show us some new properties and relations of the exponential, trigamma, the first kind modified Bessel functions and the hypergeometric series.
Keywords: property; connection; completely monotonic function; completely monotonic degree; integral representation; difference;exponential function; trigamma function; hypergeometric series; inequality; modified Bessel function.
References
M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 9th printing, Washington,1970.
B.-N. Guo and F. Qi, A completely monotonic function involving the tri-gamma function and with degree one, Appl. Math. Comput. 218 (2012), no. 19, 9890-9897; Available online at http://dx.doi.org/10.1016/j.amc.2012.03.075.
B.-N. Guo and F. Qi, Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 1007-1015; Available online at http://dx.doi.org/10.1090/S0002-9939-2012-11387-5.
B.-N. Guo and F. Qi, Some integral representations and properties of Lah numbers, available online at http://arxiv.org/abs/1402.2367.
S. Koumandos and H. L. Pedersen, Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function, J. Math. Anal. Appl. 355 (2009), no. 1, 33-40; Available online at http://dx.doi.org/10.1016/j.jmaa.2009.01.042.
F. Qi, An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions, available online at http://arxiv.org/abs/1402.2361.
F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, available online at http://arxiv.org/abs/1302.6731.
F. Qi and C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math. 10 (2013), no. 4, 1685-1696; Available online at http://dx.doi.org/10.1007/s00009-013-0272-2.
F. Qi and S.-H. Wang, Complete monotonicity of a difference between the exponential and trigamma functions and completely monotonic degree of the exponential function, available online at http://arxiv.org/abs/1210.2012.
F. Qi and X.-J. Zhang, Complete monotonicity of a difference between the exponential and trigamma functions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 21 (2014), no. 2, 141-145; Available online at http://dx.doi.org/10.7468/jksmeb.2014.21.2.141.
D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
X.-J. Zhang, Integral Representations, Properties, and Applications of Three Classes of Functions, Thesis supervised by Professor Feng Qi and submitted for the Degree of Master of Science at Tianjin Polytechnic University in January 2013. (Chinese)
X.-J. Zhang, F. Qi, and W.-H. Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, Int. J. Open Probl. Comput. Sci. Math. 5 (2012), no. 3, 122-127.
Downloads
Additional Files
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal''s published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).