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A necessary and sufficient condition for sequences to be minimal completely monotonic

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

$$ (-1)^{k}\Delta ^{k}\mu _{n} \ge 0,\quad n,k\in \mathbb{N}_{0}:=\{0\} \cup \mathbb{N}, $$
(1)

where

$$ \Delta ^{0}\mu _{n}=\mu _{n} $$
(2)

and

$$ \Delta ^{k+1}\mu _{n}=\Delta ^{k}\mu _{n+1}-\Delta ^{k}\mu _{n}. $$
(3)

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 sub-class 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. (1)

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

  2. (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

$$ \bigl\{ \mu ^{*}_{0}, \mu _{1}, \mu _{2}, \ldots \bigr\} $$
(4)

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

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{m+1} $$

converges and

$$ \mu _{m}\geq \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{m+1}. $$
(5)

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 \(n\in \mathbb{N}_{0}\)

$$ (-1)^{n}f^{(n)}(x)\geq 0, \quad x\in I^{o}. $$
(6)

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, [13, 519, 2124].

For sequences to be interpolated by completely monotonic functions, Widder [25] proved that there exists a function

$$ f\in \mathit{CM}[0,\infty ) $$

such that

$$ f(n)=\mu _{n},\quad n\in \mathbb{N}_{0} $$

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

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1} $$
(7)

converges. Let

$$ \mu _{0}^{*}:= \sum _{j=0}^{\infty }(-1)^{j}\Delta ^{j}\mu _{1}. $$
(8)

Then the sequence

$$ \bigl\{ \mu _{0}^{*},\mu _{1},\mu _{2},\mu _{3},\ldots \bigr\} $$
(9)

is minimal completely monotonic.

Remark 5

It should be noted that the condition: “the series

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1} $$
(10)

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

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1}. $$

For example, let

$$ \mu _{n}=\frac{1}{n},\quad n\in \mathbb{N}. $$

We can verify that the sequence \(\{\mu _{n}\}_{n=1}^{\infty }\) is completely monotonic and that

$$ \Delta ^{j}\mu _{1}=\frac{(-1)^{j}}{j+1}. $$

Hence

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1}= \sum_{j=0}^{\infty } \frac{1}{j+1}, $$

which is divergent.

Theorem 6

Suppose that the sequence \(\{\mu _{n}\}_{n=0}^{\infty }\) is minimal completely monotonic. Then the series

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1} $$
(11)

converges and

$$ \mu _{0}= \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1}. $$
(12)

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

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1} $$
(13)

converges, and

$$ \mu _{0}= \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1}. $$
(14)

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

$$ \bigl\{ \mu _{0}^{*},\mu _{1},\mu _{2},\mu _{3},\ldots \bigr\} $$
(15)

is completely monotonic. By Theorem 9 in [18], if a sequence

$$ \{\mu _{0},\mu _{1},\mu _{2},\mu _{3},\ldots \} $$
(16)

is completely monotonic, then

$$ \mu _{0}\geq \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1}=\mu _{0}^{*}. $$
(17)

Hence by the definition of minimal completely monotonic sequence, we know that the sequence

$$ \bigl\{ \mu _{0}^{*},\mu _{1},\mu _{2},\mu _{3},\ldots \bigr\} $$
(18)

is minimal completely monotonic. The proof of Theorem 4 is completed. □

Proof of Theorem 6

Since the sequence

$$ \{\mu _{0},\mu _{1},\mu _{2},\mu _{3},\ldots \} $$
(19)

is completely monotonic, by Theorem 9 in [18], the series

$$ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1} $$
(20)

converges and

$$ \mu _{0}\geq \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1}. $$
(21)

By Theorem 11 in [18], we see that the sequence

$$ \Biggl\{ \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1},\mu _{1}, \mu _{2},\mu _{3},\ldots \Biggr\} $$
(22)

is completely monotonic. Since the completely monotonic sequence

$$ \{\mu _{0},\mu _{1},\mu _{2},\mu _{3},\ldots \} $$
(23)

is minimal, we have

$$ \mu _{0}\leq \sum_{j=0}^{\infty }(-1)^{j} \Delta ^{j}\mu _{1}. $$
(24)

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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Contributions

All the authors contributed to the writing of the present article. They also read and approved the final manuscript.

Corresponding author

Correspondence to Xi-Feng Wang.

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The authors declare that they have no competing interests.

Additional information

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/s13662-020-03051-8

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MSC

  • 44A60
  • 44A10

Keywords

  • Completely monotonic sequence
  • Completely monotonic function
  • Minimal completely monotonic sequence