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A necessary and sufficient condition for sequences to be minimal completely monotonic
Advances in Difference Equations volume 2020, Article number: 665 (2020)
Abstract
In this article, we present a necessary and sufficient condition under which sequences are minimal completely monotonic.
1 Introduction and the main results
We first recall some definitions and basic results on completely monotonic sequences and minimal completely monotonic sequences.
Definition 1
([20])
A sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) is called completely monotonic if
where
and
Here in Definition 1, and throughout the paper, \(\mathbb{N}\) is the set of all positive integers and \(\mathbb{N}_{0}\) is the set of all nonnegative integers.
Widder [25] defined a subclass of the class of completely monotonic sequences.
Definition 2
A sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) is called minimal completely monotonic if it is completely monotonic and if it will not be completely monotonic when \(\mu _{0}\) is replaced by a number less than \(\mu _{0}\).
Regarding the relationships between completely monotonic sequences and minimal completely monotonic sequences, in [6] the author proved that if the sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) is completely monotonic, then:

(1)
for any \(m\in \mathbb{N}\), the sequence \(\{\mu _{n}\}_{n=m}^{\infty }\) is minimal completely monotonic, and

(2)
there exists one (then only one) number \(\mu ^{*}_{0}\) such that the sequence
$$ \bigl\{ \mu ^{*}_{0}, \mu _{1}, \mu _{2}, \ldots \bigr\} $$is minimal completely monotonic.
Please note that the complete monotonicity of the sequence \(\{\mu _{n}\}_{n=1}^{\infty }\) cannot guarantee that there exists a number \(\mu ^{*}_{0}\) such that the sequence
is completely monotonic. In fact, if the sequence (4) is completely monotonic, then the sequence \(\{\mu _{n}\}_{n=1}^{\infty }\) should be minimal completely monotonic.
In [18] the authors showed that if the sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) is completely monotonic, then, for any \(m\in \mathbb{N}_{0}\), the series
converges and
We also recall the following definition.
Definition 3
([4])
A function f is said to be completely monotonic on an interval I, if \(f \in C(I)\), has derivatives of all orders on \(I^{o}\) (the interior of I) and for all \(n\in \mathbb{N}_{0}\)
Here in Definition 3\(C(I)\) is the space of all continuous functions on the interval I. The class of all completely monotonic functions on the interval I is denoted by \(\mathit{CM}(I)\).
There is rich literature on completely monotonic functions and sequences, and their applications. For more recent works, see, for example, [1–3, 5–19, 21–24].
For sequences to be interpolated by completely monotonic functions, Widder [25] proved that there exists a function
such that
if and only if the sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) is minimal completely monotonic. From this we see that the condition of minimal complete monotonicity is critical for a sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) to be interpolated by a completely monotonic function on the interval \([0,\infty )\).
In this article, we shall further investigate on minimal completely monotonic sequences. The main results of this article are as follows.
Theorem 4
Suppose that the sequence \(\{\mu _{n}\}_{n=1}^{\infty }\) is completely monotonic and that the series
converges. Let
Then the sequence
is minimal completely monotonic.
Remark 5
It should be noted that the condition: “the series
converges” in Theorem 4 cannot be dropped since the complete monotonicity of the sequence \(\{\mu _{n}\}_{n=1}^{\infty }\) cannot guarantee the convergence of the series
For example, let
We can verify that the sequence \(\{\mu _{n}\}_{n=1}^{\infty }\) is completely monotonic and that
Hence
which is divergent.
Theorem 6
Suppose that the sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) is minimal completely monotonic. Then the series
converges and
Theorem 7
A necessary and sufficient condition for the sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) to be minimal completely monotonic is that the sequence \(\{\mu _{n}\}_{n=1}^{\infty }\) is completely monotonic, the series
converges, and
2 Proof of the main results
Now we are in a position to prove the main results.
Proof of Theorem 4
By Theorem 11 in [18], we see that the sequence
is completely monotonic. By Theorem 9 in [18], if a sequence
is completely monotonic, then
Hence by the definition of minimal completely monotonic sequence, we know that the sequence
is minimal completely monotonic. The proof of Theorem 4 is completed. □
Proof of Theorem 6
Since the sequence
is completely monotonic, by Theorem 9 in [18], the series
converges and
By Theorem 11 in [18], we see that the sequence
is completely monotonic. Since the completely monotonic sequence
is minimal, we have
From (21) and (24), we get our conclusion. The proof of Theorem 6 is completed. □
Proof of Theorem 7
By the definition of completely monotonic sequence, Theorem 9 in [18] and Theorem 6, we know that the condition is necessary. By Theorem 4, we see that the condition is sufficient. The proof of Theorem 7 is thus completed. □
3 Conclusion
In this paper, we investigated properties of completely monotonic sequences. We have proved a necessary condition for a sequence to be a minimal completely monotonic sequence. We also have presented a necessary and sufficient condition under which sequences are minimal completely monotonic.
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Acknowledgements
The authors thank the editor and the reviewers for their valuable suggestions and comments which have improved the manuscript significantly.
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The present investigation was supported, in part, by the Natural Science Foundation of China under Grant 11401604.
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Dedicated to Professor Hari M. Srivastava on the occasion of his eightieth birthday.
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Wang, XF., Ismail, M.E.H., Batir, N. et al. A necessary and sufficient condition for sequences to be minimal completely monotonic. Adv Differ Equ 2020, 665 (2020). https://doi.org/10.1186/s13662020030518
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DOI: https://doi.org/10.1186/s13662020030518
MSC
 44A60
 44A10
Keywords
 Completely monotonic sequence
 Completely monotonic function
 Minimal completely monotonic sequence