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 R^{n} 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 |
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Copyright © The Author(s), 2020. Published by Science Publishing Group |
Completely Monotonic, Inequality, Logarithmically Completely Monotonic Function, Gamma Function
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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
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
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
@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} }
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 -