Complete Monotonicity of Functions Connected with the Exponential Function and Derivatives

Abstract

Some complete monotonicity results that the functions are logarithmically completely monotonic, and that differences between consecutive derivatives of these two functions are completely monotonic, and that the ratios between consecutive derivatives of these two functions are decreasing on are discovered. As applications of these newly discovered results, some complete monotonicity results concerning the polylogarithm are found. Finally a conjecture on the complete monotonicity of the above-mentioned ratios is posed.

1. Introduction and Main Results

Throughout this paper, we denote the set of all positive integers by .

Recently, the following problem was posed in [1, page 569]. For and , determine the numbers for such that This problem, among other things, was answered in [1] by eight identities. Two of the eight identities may be recited as follows.

Theorem A (see [1, Theorem 2.1]). For , one has where are Stirling numbers of the second kind.

Theorem B (see [1, Theorem 2.3]). For with , the coefficients defined in (1) may be calculated by where

For more information, please refer to [2, 3] and related references therein.

Stimulated by results obtained in [1], as mentioned above, we are interested in two functional sequences: where and , and we firstly discover in this paper the following results.

Theorem 1. For , the functions and are completely monotonic on . More strongly, the functions and are logarithmically completely monotonic on .
For , the functions and satisfy

Theorem 2. For given , the differences are completely monotonic functions on . In particular, the sequences and are increasing with respect to for ; that is, the inequalities are valid for all and .

Theorem 3. For given , the ratios are decreasing on , 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 and defined in (10).

2. 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.

Definition 4 (see [4, 5]). A function is said to be completely monotonic on an interval if has derivatives of all orders on and for and .
The noted Hausdorff-Bernstein-Widder theorem [5, page 161, Theorem 12b] says that a necessary and sufficient condition that should be completely monotonic for is that where is nondecreasing and the integral converges for . In other words, a function defined on is completely monotonic on 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 if it has derivatives of all orders on and its logarithm satisfies for on .
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 [7–9] 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 and that is a sequence of positive and differentiable functions such that the series converge absolutely and uniformly over compact subsets of .(1)If the logarithmic derivative forms an increasing sequence of functions and if decreases (resp., increases), then decreases (resp., increases) for .(2)If the logarithmic derivative forms a decreasing sequence of functions and if decreases (resp., increases), then the function increases (resp., decreases) for .

3. Proofs of Main Results

Now we start out to prove our theorems.

Proof of Theorem 1. Since for , the functions and for are completely monotonic on .
Taking the logarithms of the functions and and differentiating yield Consequently, by definition of logarithmically completely monotonic functions and the above obtained complete monotonicity of the function , it is ready to deduce the logarithmically complete monotonicity of and on .
The formulas in (7) can be straightforwardly verified. The proof of Theorem 1 is complete.

Proof of Theorem 2. A simple computation yields for . By definition, the difference for is completely monotonic on .
From the first equalities in (15), respectively, it follows that the functions are all completely monotonic on .
The inequalities in (9) follow from the positivity of and for . The proof of Theorem 2 is complete.

Proof of Theorem 3. It is easy to see that Let , , and . Then is an increasing sequence and form a decreasing sequence. By Lemma 6, it follows that the function is decreasing on .
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 for . Accordingly, the function has the same monotonicity and the same limit for as does for . As a result, the function is decreasing on and the third limit in (11) is valid for .
A straightforward computation gives which obviously tends to as and apparently decreases on . The proof of Theorem 3 is complete.

4. Some Applications

We recall from [12] that the polylogarithm is the function defined for all and over the open unit disk in the complex plane . Its definition on the whole complex plane then follows uniquely via analytic continuation.

For given , since , we observe that Therefore, considering (15), Theorems 1 to 3 can be used to find some properties of the polylogarithm as follows.

Theorem 7. For , the polylogarithm is completely monotonic with respect to . More strongly, the polylogarithm is logarithmically completely monotonic with respect to .
For , the polylogarithm satisfies

Theorem 8. For given , the differences are completely monotonic functions on . In particular, the polylogarithm sequence is increasing with respect to for given ; that is, the inequality is valid for all and .

Theorem 9. For given , the ratios are decreasing on , 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 for as follows.

Theorem 10. The following complete monotonicity is valid.(1)The polylogarithm is logarithmically completely monotonic with respect to .(2)For , the polylogarithm is completely monotonic with respect to .(3)For given , the differences are completely monotonic functions on .

Proof. The second item of [13, Theorem 5] tells us that if is completely monotonic on an interval and is logarithmically completely monotonic on the domain , then the composite function is logarithmically completely monotonic on . It is easy to verify that the derivative of is and completely monotonic on . Combining these conclusions with the logarithmically complete monotonicity of in Theorem 7 yields that the polylogarithm is logarithmically completely monotonic with respect to .
In [14, page 83], it was given that if and are functions such that is defined on and if and are completely monotonic, then is also completely monotonic on . Replacing by and by and making use of the complete monotonicity of in Theorem 7 lead to the complete monotonicity of the polylogarithm .
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.

5. A Conjecture

By Theorem 1 and the fact that any logarithmically completely monotonic function on is also completely monotonic on , it is clear that the functions are all completely monotonic on . This motivates us to pose the following conjecture.

Conjecture 12. For given , the functions and defined in (10) or (27) are completely monotonic on .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments on and careful corrections to the original version of this paper. The authors appreciate Professor Dr. Feng Qi in China for his valuable contribution to this paper. Chun-Fu Wei was partially supported by the NNSF under Grant no. 51274086 of China, by the Ministry of Education Doctoral Foundation of China—Priority Areas under Grant no. 20124116130001, and by the State Key Laboratory Cultivation Base for Gas Geology and Gas Control under Grant no. WS2012A10 at Henan Polytechnic University, China.