Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions

Mohammad Soueycatt; Abedalbaset Yonsoo; Ahmad Bekdash; Nabil Khuder Salman

doi:doi:10.11648/j.ajam.20200801.13

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Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions

Mohammad Soueycatt, Abedalbaset Yonsoo, Ahmad Bekdash, Nabil Khuder Salman

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.

DOI	10.11648/j.ajam.20200801.13
Published in	American Journal of Applied Mathematics (Volume 8, Issue 1, February 2020)
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), 2024. Published by Science Publishing Group

Keywords

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

References

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[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.
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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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TY  - JOUR
T1  - Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions
AU  - Mohammad Soueycatt
AU  - Abedalbaset Yonsoo
AU  - Ahmad Bekdash
AU  - Nabil Khuder Salman
Y1  - 2020/01/31
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200801.13
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
SP  - 17
EP  - 21
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200801.13
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.
VL  - 8
IS  - 1
ER  -

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Author Information

Mohammad Soueycatt

Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria
Abedalbaset Yonsoo

Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria
Ahmad Bekdash

Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria
Nabil Khuder Salman

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

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions
AU  - Mohammad Soueycatt
AU  - Abedalbaset Yonsoo
AU  - Ahmad Bekdash
AU  - Nabil Khuder Salman
Y1  - 2020/01/31
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200801.13
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
SP  - 17
EP  - 21
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200801.13
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.
VL  - 8
IS  - 1
ER  -

Copy | Download