A certain class of completely monotonic sequences
Advances in Difference Equations volume 2013, Article number: 294 (2013)
In this article, we present some necessary conditions, a sufficient condition and a necessary and sufficient condition for sequences to be completely monotonic. One counterexample is also presented.
MSC:40A05, 26A45, 26A48, 39A60.
1 Introduction and the main results
We first recall some definitions and basic results on or related to completely monotonic sequences and completely monotonic functions.
Definition 1 
A sequence is called a moment sequence if there exists a function of bounded variation on the interval such that
Here, in Definition 1 and throughout the paper,
and ℕ is the set of all positive integers.
Definition 2 
A sequence is called completely monotonic if
Such a sequence is called totally monotone in .
From Definition 2, using mathematical induction, we can prove, for a completely monotonic sequence , that the sequence is non-increasing for any fixed , and that the sequence is non-increasing for any fixed . The difference equation (4) plays an important role in the proofs of these properties and our main results of this paper.
In , the authors showed that for a completely monotonic sequence , we always have
unless , a constant for all .
It was shown (see ) as follows.
Theorem 1 A sequence is a moment sequence if and only if there exists a constant L such that
where in (7), is defined by (6).
For completely monotonic sequences, the following is the well-known Hausdorff’s theorem (see ).
Theorem 2 A sequence is completely monotonic if and only if there exists a non-decreasing and bounded function on such that
From this theorem, we know (see ) that a completely monotonic sequence is a moment sequence and is as follows.
Theorem 3 A necessary and sufficient condition that the sequence should be a moment sequence is that it should be the difference of two completely monotonic sequences.
We also recall the following definition.
Definition 3 
A function f is said to be completely monotonic on an interval I if f is continuous on I has derivatives of all orders on (the interior of I) and for all ,
Some mathematicians use the terminology completely monotone instead of completely monotonic. The class of all completely monotonic functions on the interval I is denoted by .
For the completely monotonic functions on the interval , Widder proved (see ).
Theorem 4 A function f on the interval is completely monotonic if and only if there exists a bounded and non-decreasing function on such that
There exists a close relationship between completely monotonic functions and completely monotonic sequences. For example, Widder  showed the following.
Theorem 5 Suppose that , then for any , the sequence is completely monotonic.
This result was generalized in  as follows.
Theorem 6 Suppose that . If the sequence is completely monotonic and , then the sequence is also completely monotonic.
For the meaning of , in Theorem 6, see (3) and (4).
Suppose that . By Theorem 5, we know that is completely monotonic.
The following result was obtained in .
Theorem 7 Suppose that the sequence is completely monotonic, then for any , there exists a continuous interpolating function on the interval such that and are both completely monotonic and
From this result or Theorem 2, we can get the following.
Theorem 8 Suppose that the sequence is completely monotonic. Then there exists a completely monotonic interpolating function on the interval such that
It should be noted that (see [, Chapter IV]) under the condition of Theorem 8, we cannot guarantee that there exists a completely monotonic interpolating function on the interval such that
In this article, we shall further investigate the properties of the completely monotonic sequences. We shall give some necessary conditions, a sufficient condition and a necessary and sufficient condition for sequences to be completely monotonic. More precisely we have the following results.
Theorem 9 Suppose that the sequence is completely monotonic. Then, for any , the series
Corollary 1 Suppose that the sequence is completely monotonic. Then for ,
Remark 1 Although from the complete monotonicity of the sequence , we can deduce that for any , the series
converges, it cannot guarantee the convergence of the series
For example, let
Since the function
is completely monotonic on the interval , by Theorem 5, we see that the sequence
is completely monotonic. This conclusion can also be obtained by setting
in Theorem 2.
We can verify that
is the famous harmonic series, which is divergent.
Theorem 10 Suppose that the sequence is completely monotonic. Then for any ,
Theorem 11 Suppose that the sequence is completely monotonic and that the series
converges. Let be such that
Then the sequence is completely monotonic.
Theorem 12 A necessary and sufficient condition for the sequence to be completely monotonic is that the sequence is completely monotonic, the series
2 Proofs of the main results
Now, we are in a position to prove the main results.
Proof of Theorem 9 Since is completely monotonic, by Theorem 2, there exists a non-decreasing and bounded function on the interval such that
From (3), (4) and (13), we can prove that
Now, for , we have
Hence, for ,
from (15), we get, for ,
From (16), we also know that
is a positive series. Then by (17), we obtain that
converges and that
The proof of Theorem 9 is thus completed. □
Proof of Corollary 1 This corollary can be obtained from (15). □
Proof of Theorem 10 Let m be a fixed non-negative integer.
From Theorem 9, we see that
which means that (12) is valid for .
Suppose that (12) is valid for . Then
which means that (12) is valid for . Therefore, by mathematical induction, (12) is valid for all . The proof of Theorem 10 is completed. □
Proof of Theorem 11 By the definition of completely monotonic sequence, we only need to prove that
We first prove that
From the condition of Theorem 11, (22) is valid for .
Suppose that (22) is valid for . Then we have
which means that (22) is valid for . Therefore, by mathematical induction, (22) is valid for all .
is a convergent positive series, we know that
From (22) and (24), we obtain that
The proof of Theorem 11 is completed. □
Proof of Theorem 12 By Definition 2 and by setting in Theorem 9, we see that the condition is necessary. By Theorem 11, we know that the condition is sufficient. The proof of Theorem 12 is completed. □
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The authors thank the editor and the referees, one of whom brought our attention to the reference , for their valuable suggestions to improve the quality of this paper. The present investigation was supported, in part, by the Natural Science Foundation of Henan Province of China under Grant 112300410022.
The authors declare that they have no competing interests.
All the authors contributed to the writing of the present article. They also read and approved the final manuscript.
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Guo, S., Srivastava, H.M. & Batir, N. A certain class of completely monotonic sequences. Adv Differ Equ 2013, 294 (2013). https://doi.org/10.1186/1687-1847-2013-294