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

$$ (-1)^{k}\Delta ^{k}\mu _{n} \ge 0,\quad n,k\in \mathbb{N}_{0}:=\{0\} \cup \mathbb{N}, $$
(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, [1–3, 5–19, 21–24].

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)

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

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

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.