Complete Monotonicity of Functions Connected with the Exponential Function and Derivatives

and Applied Analysis 3 but not conversely. For more information on this class of functions, please refer to [7–9] and [10, Section 1.3] and closely related references therein.This shows that it is helpful to prove the logarithmically complete monotonicity of functions. Lemma 6 (see [11, Lemma 2.2]). Suppose that a k , b k > 0 and that {u k (t)} is a sequence of positive and differentiable functions such that the series


Introduction and Main Results
Throughout this paper, we denote the set of all positive integers by N.
This problem, among other things, was answered in [1] by eight identities. Two of the eight identities may be recited as follows.
Stimulated by results obtained in [1], as mentioned above, we are interested in two functional sequences: where ∈ (0, ∞) and ∈ {0} ∪ N, and we firstly discover in this paper the following results.

Theorem 3.
For given ∈ {0} ∪ N, the ratios are decreasing on (0, ∞), with In order to further show the importance and significance of the two functional sequences in (6), we secondly give several applications of Theorems 1 to 3 in Section 4.
Finally, we pose a conjecture on the complete monotonicity of the functions F ( ) and G ( ) defined in (10).

Two Definitions and a Lemma
Now we list definitions of the completely monotonic and the logarithmically completely monotonic functions, which just now appeared in Theorems 1 and 2, and recite a lemma, which is needed to prove Theorem 3.
The noted Hausdorff-Bernstein-Widder theorem [5, page 161, Theorem 12b] says that a necessary and sufficient condition that ( ) should be completely monotonic for 0 < < ∞ is that where ( ) is nondecreasing and the integral converges for 0 < < ∞. In other words, a function defined on (0, ∞) is completely monotonic on (0, ∞) if and only if it is a Laplace transform. For more information on the theory of completely monotonic functions, please refer to [4, Chapter XIII], [5, Chapter IV], and the newly published monograph [6]. This means that it is useful to confirm the complete monotonicity of functions.
Definition 5 (see [7,8]). A positive function ( ) is said to be logarithmically completely monotonic on an interval ⊆ R if it has derivatives of all orders on and its logarithm ln ( ) It has been proved that any logarithmically completely monotonic function on is also completely monotonic on , but not conversely. For more information on this class of functions, please refer to [7][8][9] and [10, Section 1.3] and closely related references therein. This shows that it is helpful to prove the logarithmically complete monotonicity of functions.
Lemma 6 (see [11,Lemma 2.2]). Suppose that , > 0 and that { ( )} is a sequence of positive and differentiable functions such that the series converge absolutely and uniformly over compact subsets of [0, ∞).

Proofs of Main Results
Now we start out to prove our theorems.
Taking the logarithms of the functions 0 ( ) and 0 ( ) and differentiating yield Consequently, by definition of logarithmically completely monotonic functions and the above obtained complete monotonicity of the function 0 ( ), it is ready to deduce the logarithmically complete monotonicity of 0 ( ) and 0 ( ) on (0, ∞).
The formulas in (7) can be straightforwardly verified. The proof of Theorem 1 is complete.

Proof of Theorem 2. A simple computation yields
for ∈ N. By definition, the difference F ( ) = +1 ( ) − ( ) for ∈ {0} ∪ N is completely monotonic on (0, ∞). From the first equalities in (15), respectively, it follows that the functions are all completely monotonic on (0, ∞). The inequalities in (9) follow from the positivity of F ( ) and G ( ) for ∈ N. The proof of Theorem 2 is complete.
Taking → ∞ at the very ends of (19) shows that The first limit in (11) thus follows. Making use of the second relation in (15) yields G ( ) = F ( ) for ∈ N. Accordingly, the function G ( ) has the same monotonicity and the same limit for → ∞ as F ( ) does for ∈ N. As a result, the function G ( ) is decreasing on (0, ∞) and the third limit in (11) is valid for ∈ N.
A straightforward computation gives which obviously tends to 0 as → ∞ and apparently decreases on (0, ∞). The proof of Theorem 3 is complete.

Some Applications
We recall from [12] that the polylogarithm Li ( ) is the function defined for all ∈ C and over the open unit disk | | < 1 in the complex plane C. Its definition on the whole complex plane then follows uniquely via analytic continuation. For given ∈ {0}∪N, since ( ) = ∑ ∞ ℓ=1 ℓ −ℓ , we observe that Therefore, considering (15), Theorems 1 to 3 can be used to find some properties of the polylogarithm Li ( ) as follows.

Theorem 9.
For given ∈ {0} ∪ N, the ratios are decreasing on (0, ∞), with Furthermore, by the above (logarithmically) complete monotonicity in Theorems 7 and 8 and by some complete monotonicity properties of composite functions, we can obtain the complete monotonicity of functions involving the polylogarithm Li − (1/ ) for ∈ {0} ∪ N as follows.
Theorem 10. The following complete monotonicity is valid.
Proof. The second item of [13,Theorem 5] tells us that if ℎ ( ) is completely monotonic on an interval and ( ) is logarithmically completely monotonic on the domain ℎ( ), then the composite function ∘ ℎ( ) = (ℎ( )) is logarithmically completely monotonic on . It is easy to verify that the derivative of ln is 1/ and completely monotonic on (0, ∞). Combining these conclusions with the logarithmically complete monotonicity of Li 0 ( − ) in Theorem 7 yields that the polylogarithm Li 0 (1/ ) = 1/( − 1) is logarithmically completely monotonic with respect to ∈ (1, ∞).
In [14, page 83], it was given that if and are functions such that ( ( )) is defined on (0, ∞) and if and are completely monotonic, then → ( ( )) is also completely monotonic on (0, ∞). Replacing ( ) by Li − ( − ) and ( ) by ln and making use of the complete monotonicity of Li − ( − ) in Theorem 7 lead to the complete monotonicity of the polylogarithm Li − (1/ ).
The leftover proofs are similar to the above arguments. The proof of Theorem 10 is complete.
Remark 11. To the best of our knowledge, the above (logarithmically) complete monotonicity results concerning the polylogarithm are new. This shows us the importance of the two functional sequences in (6) and the significance of Theorems 1 to 3.