Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions
American Journal of Applied Mathematics
Volume 8, Issue 1, February 2020, Pages: 17-21
Received: Nov. 29, 2019;
Accepted: Dec. 21, 2019;
Published: Jan. 31, 2020
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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
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.
Nabil Khuder Salman,
Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions, American Journal of Applied Mathematics.
Vol. 8, No. 1,
2020, pp. 17-21.
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