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Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions

Received: 29 November 2019     Accepted: 21 December 2019     Published: 31 January 2020
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Abstract

The monotonic functions were first introduced by S. Bernstein as functions which are non-negative with non-negative derivatives of all orders. He proved that such functions are necessarily analytic and he showed later that if a function is absolutely monotonic on the negative real axis then it can be represented there by a Laplace- Stieitjes integral with non-decreasing determining function and converse. Somewhat earlier F. Hausdorff had proved a similar result for completely monotonic sequences which essentially contained the Bernstein result. Bernstein was evidently unaware of Hausdorff's result, and his proof followed entirely independent lines. Since then many studies have been written on monotonic functions. In this work, we mainly have proved that a certain function involving ratio of the Euler gamma functions and some parameters is completely and logarithmically completely monotonic. Also, we have given the sufficient conditions for this function to be respectively completely and logarithmically completely monotonic. As applications, some inequalities involving the volume of the unite ball in the Euclidian space Rn are obtained. The established results not only unify and improve certain known inequalities including, but also can generate some new inequalities and the given results could trigger a new research direction in the theory of inequalities and special functions.

Published in American Journal of Applied Mathematics (Volume 8, Issue 1)
DOI 10.11648/j.ajam.20200801.13
Page(s) 17-21
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Completely Monotonic, Inequality, Logarithmically Completely Monotonic Function, Gamma Function

References
[1] D. V. Widder, The Laplace transform. New York: Princeton University Press, 1941.
[2] A. Grinshpan and M. Ismail, “Completely monotonic functions involving the gamma and q-gamma functions,” Proc. Am. Math. Soc., vol. 134, no. 4, pp. 1153–1160, 2006.
[3] R. A. Horn, “On infinitely divisible matrices, kernels, and functions,” Z. Für Wahrscheinlichkeitstheorie Verwandte Geb., vol. 8, no. 3, pp. 219–230, 1967.
[4] H. Alzer and C. Berg, “Some classes of completely monotonic functions, II,” Ramanujan J., vol. 11, no. 2, pp. 225–248, 2006.
[5] C. Berg, “Integral Representation of Some Functions Related to the Gamma Function,” Mediterr. J. Math., vol. 1, no. 4, pp. 433–439, 2004.
[6] F. Qi, and B.-N. Guo, “Logarithmically complete monotonicity of Catalan-Qi function related to Catalan numbers,” Cogent Math., vol. 3, no. 1, pp. 1–6, 2016.
[7] K. Nantomah and L. Yin, “Logarithmically Complete Monotonicity of Certain Ratios Involving the $k$-Gamma Function,” 2019.
[8] F. Qi and A.-Q. Liu, “Completely monotonic degrees for a difference between the logarithmic and psi functions,” J. Comput. Appl. Math., vol. 361, pp. 366–371, 2019.
[9] F. Qi, “Integral representations for multivariate logarithmic polynomials,” J. Comput. Appl. Math., vol. 336, pp. 54–62, 2018.
[10] J. El Kamel and K. Mehrez, “A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions,” Positivity, vol. 22, no. 5, pp. 1403–1417, 2018.
[11] L. Yin, L.-G. Huang, X.-L. Lin, and Y.-L. Wang, “Monotonicity, concavity, and inequalities related to the generalized digamma function,” Adv. Differ. Equ., vol. 2018, no. 1, p. 246, 2018.
[12] L. Yin, L.-G. Huang, Z.-M. Song, and X. K. Dou, “Some monotonicity properties and inequalities for the generalized digamma and polygamma functions,” J. Inequalities Appl., vol. 2018, no. 249, pp. 1–13, 2018.
[13] B.-N. Guo and F. Qi, “Inequalities and monotonicity for the ratio of gamma functions,” Taiwan. J. Math., vol. 7, no. 2, pp. 239–247, 2003.
[14] F. Qi and B.-N. Guo, “A logarithmically completely monotonic function involving the gamma function,” Taiwan. J. Math., vol. 14, no. 4, pp. 1623–1628, 2010.
[15] T.-H. Zhao, Y.-M. Chu, and Y.-P. Jiang, “Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions,” J. Inequalities Appl., vol. 2009, no. 1, p. 728612, 2009.
[16] Y. Yu, “An inequality for ratios of gamma functions,” J. Math. Anal. Appl., vol. 352, no. 2, pp. 967–970, Apr. 2009.
[17] B.-N. Guo and F. Qi, “An extension of an inequality for ratios of gamma functions,” J. Approx. Theory, vol. 163, no. 9, pp. 1208–1216, 2011.
[18] F. Qi and B.-N. Guo, “Two new proofs of the complete monotonicity of a function involving the psi function,” Bull. Korean Math. Soc., vol. 47, no. 1, pp. 103–111, 2010.
Cite This Article
  • APA Style

    Mohammad Soueycatt, Abedalbaset Yonsoo, Ahmad Bekdash, Nabil Khuder Salman. (2020). Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions. American Journal of Applied Mathematics, 8(1), 17-21. https://doi.org/10.11648/j.ajam.20200801.13

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    ACS Style

    Mohammad Soueycatt; Abedalbaset Yonsoo; Ahmad Bekdash; Nabil Khuder Salman. Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions. Am. J. Appl. Math. 2020, 8(1), 17-21. doi: 10.11648/j.ajam.20200801.13

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    AMA Style

    Mohammad Soueycatt, Abedalbaset Yonsoo, Ahmad Bekdash, Nabil Khuder Salman. Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions. Am J Appl Math. 2020;8(1):17-21. doi: 10.11648/j.ajam.20200801.13

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  • @article{10.11648/j.ajam.20200801.13,
      author = {Mohammad Soueycatt and Abedalbaset Yonsoo and Ahmad Bekdash and Nabil Khuder Salman},
      title = {Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions},
      journal = {American Journal of Applied Mathematics},
      volume = {8},
      number = {1},
      pages = {17-21},
      doi = {10.11648/j.ajam.20200801.13},
      url = {https://doi.org/10.11648/j.ajam.20200801.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200801.13},
      abstract = {The monotonic functions were first introduced by S. Bernstein as functions which are non-negative with non-negative derivatives of all orders. He proved that such functions are necessarily analytic and he showed later that if a function is absolutely monotonic on the negative real axis then it can be represented there by a Laplace- Stieitjes integral with non-decreasing determining function and converse. Somewhat earlier F. Hausdorff had proved a similar result for completely monotonic sequences which essentially contained the Bernstein result. Bernstein was evidently unaware of Hausdorff's result, and his proof followed entirely independent lines. Since then many studies have been written on monotonic functions. In this work, we mainly have proved that a certain function involving ratio of the Euler gamma functions and some parameters is completely and logarithmically completely monotonic. Also, we have given the sufficient conditions for this function to be respectively completely and logarithmically completely monotonic. As applications, some inequalities involving the volume of the unite ball in the Euclidian space Rn are obtained. The established results not only unify and improve certain known inequalities including, but also can generate some new inequalities and the given results could trigger a new research direction in the theory of inequalities and special functions.},
     year = {2020}
    }
    

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    AU  - Mohammad Soueycatt
    AU  - Abedalbaset Yonsoo
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    DO  - 10.11648/j.ajam.20200801.13
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    AB  - The monotonic functions were first introduced by S. Bernstein as functions which are non-negative with non-negative derivatives of all orders. He proved that such functions are necessarily analytic and he showed later that if a function is absolutely monotonic on the negative real axis then it can be represented there by a Laplace- Stieitjes integral with non-decreasing determining function and converse. Somewhat earlier F. Hausdorff had proved a similar result for completely monotonic sequences which essentially contained the Bernstein result. Bernstein was evidently unaware of Hausdorff's result, and his proof followed entirely independent lines. Since then many studies have been written on monotonic functions. In this work, we mainly have proved that a certain function involving ratio of the Euler gamma functions and some parameters is completely and logarithmically completely monotonic. Also, we have given the sufficient conditions for this function to be respectively completely and logarithmically completely monotonic. As applications, some inequalities involving the volume of the unite ball in the Euclidian space Rn are obtained. The established results not only unify and improve certain known inequalities including, but also can generate some new inequalities and the given results could trigger a new research direction in the theory of inequalities and special functions.
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Author Information
  • Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria

  • Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria

  • Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria

  • Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria

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