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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.
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
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. □
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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Funding
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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MSC
 44A60
 44A10
Keywords
 Completely monotonic sequence
 Completely monotonic function
 Minimal completely monotonic sequence