Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions

The monotonic functions were first introduced by S. Bernstein as functions which are non-negative with nonnegative 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 LaplaceStieitjes integral with nondecreasing 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 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.


Introduction
A function is said to be completely monotonic [1] on interval ⊆ ℝ if it has derivatives of all orders on and satisfies for all > 0 and ∈ ℕ 0 ≤ −1 < ∞.
A positive function is said to be logarithmically completely monotonic (see for example [2]) on an interval ⊆ ℝ if it has derivatives of all orders on and its logarithm ln satisfies for ∈ ℕ 0 ≤ −1 ln < ∞.
The class of completely monotonic functions on 0, ∞ may be characterized by [1] as: is completely monotonic for 0 < < 1 if and only if where " # is non-decreasing and the integral converges for 0 < < ∞.
It is proved that is logarithmically completely monotonic if and only if % is completely monotonic for all & > 0 [3].
It is known that any logarithmically completely monotonic function must be completely monotonic, but not conversely [4].
The logarithmically completely monotonic function was characterized as the infinitely divisible completely monotonic functions [5]. Recently, the completely monotonic or logarithmically completely monotonic functions have been the subject of intensive research. For more details we refer the reader to [6]- [12].
The gamma function Γ is defined for > 0 by the integral is decreasing with respect to ≥ 1 for fixed 4 ≥ 0.
Consequently, for positive real numbers ≥ 1 and 4 ≥ 0, we have (2) The function (1) was proved to be logarithmically completely monotonic with respect to ∈ 0, ∞ for 4 ≥ 0 and so is its reciprocal for −1 < 4 ≤ − ) 5 [14]. Consequently, the inequality (2) is true for It is clear that the ranges of , 4 and & in the function ℎ %,/ extend the corresponding ones in the functions (1) and (3) which were ever discussed in [15].
In this work, we prove that the function ℎ %,/ is logarithmically completely monotonic function in some cases.

Results
Our main results are the following:  (4) is logarithmically completely monotonic with respect to ∈ −4 − 1, ∞ . c The necessary condition for the reciprocal of the function (4) to be logarithmically completely monotonic with respect to ∈ −4 − 1, ∞ is & ≤ 1. The theorem 1 extends and generalizes the logarithmically complete monotonicity of the function (1) established in [14] and a part of the results in [15].
From the theorems 1 we get the following corollary: Corollary 1. For ( > 0, 4 + 1 > 0 and + 4 + 1 > 0, the double inequality The inequality (5) generalizes and extends the inequality (2) and the main results in [16,17] For In order to prove our main results, the following lemma is needed.
Hence, the limit of the function (16)

Conclusion
We have established the necessary and sufficient conditions for a certain function involving ratio of the gamma functions to be logarithmically complete monotonicity properties. As a consequence, we derived some inequalities involving the gamma functions. The established results could trigger a new research direction in the theory of inequalities and special functions.