A certain class of completely monotonic sequences
© Guo et al.; licensee Springer. 2013
Received: 6 June 2013
Accepted: 3 September 2013
Published: 7 November 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.
Keywordsnecessary condition sufficient condition necessary and sufficient condition difference equation moment sequence completely monotonic sequence completely monotonic function bounded variation Stieltjes integral
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 
and ℕ is the set of all positive integers.
Definition 2 
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.
unless , a constant for all .
It was shown (see ) as follows.
where in (7), is defined by (6).
For completely monotonic sequences, the following is the well-known Hausdorff’s theorem (see ).
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 
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 ).
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 .
From this result or Theorem 2, we can get the following.
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.
in Theorem 2.
is the famous harmonic series, which is divergent.
Then the sequence is completely monotonic.
2 Proofs of the main results
Now, we are in a position to prove the main results.
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.
which means that (12) is valid for .
which means that (12) is valid for . Therefore, by mathematical induction, (12) is valid for all . The proof of Theorem 10 is completed. □
From the condition of Theorem 11, (22) is valid for .
which means that (22) is valid for . Therefore, by mathematical induction, (22) is valid for all .
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. □
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.
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