Theory and Modern Applications

# 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.

## 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, [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 $$[0,\infty )$$.

### 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.

## References

1. Alzer, H., Batir, N.: Monotonicity properties of the gamma function. Appl. Math. Lett. 20, 778–781 (2007)

2. Alzer, H., Berg, C., Koumandos, S.: On a conjecture of Clark and Ismail. J. Approx. Theory 134, 102–113 (2005)

3. Batir, N.: On some properties of the gamma function. Expo. Math. 26, 187–196 (2008)

4. Bernstein, S.: Sur la définition et les propriétés des fonctions analytiques d’une variable réelle. Math. Ann. 75, 449–468 (1914)

5. Guo, S.: Logarithmically completely monotonic functions and applications. Appl. Math. Comput. 221, 169–176 (2013)

6. Guo, S.: Some properties of completely monotonic sequences and related interpolation. Appl. Math. Comput. 219, 4958–4962 (2013)

7. Guo, S.: A class of logarithmically completely monotonic functions and their applications. J. Appl. Math. 2014, 757462 (2014)

8. Guo, S.: Some conditions for a class of functions to be completely monotonic. J. Inequal. Appl. 2015, 11 (2015)

9. Guo, S.: On completely monotonic and related functions. Filomat 30, 2083–2090 (2016)

10. Guo, S.: Some properties of functions related to completely monotonic functions. Filomat 31, 247–254 (2017)

11. Guo, S., Laforgia, A., Batir, N., Luo, Q.-M.: Completely monotonic and related functions: their applications. J. Appl. Math. 2014, 768516 (2014)

12. Guo, S., Qi, F.: A class of logarithmically completely monotonic functions associated with the gamma function. J. Comput. Appl. Math. 224, 127–132 (2009)

13. Guo, S., Qi, F., Srivastava, H.M.: Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic. Integral Transforms Spec. Funct. 18, 819–826 (2007)

14. Guo, S., Qi, F., Srivastava, H.M.: Supplements to a class of logarithmically completely monotonic functions associated with the gamma function. Appl. Math. Comput. 197, 768–774 (2008)

15. Guo, S., Qi, F., Srivastava, H.M.: A class of logarithmically completely monotonic functions related to the gamma function with applications. Integral Transforms Spec. Funct. 23, 557–566 (2012)

16. Guo, S., Srivastava, H.M.: A class of logarithmically completely monotonic functions. Appl. Math. Lett. 21, 1134–1141 (2008)

17. Guo, S., Srivastava, H.M.: A certain function class related to the class of logarithmically completely monotonic functions. Math. Comput. Model. 49, 2073–2079 (2009)

18. Guo, S., Srivastava, H.M., Batir, N.: A certain class of completely monotonic sequences. Adv. Differ. Equ. 2013, 294 (2013)

19. Guo, S., Srivastava, H.M., Cheung, W.S.: Some properties of functions related to certain classes of completely conotonic functions and logarithmically completely conotonic functions. Filomat 28, 821–828 (2014)

20. Hausdorff, F.: Summationsmethoden und momentfolgen I. Math. Z. 9, 74–109 (1921)

21. Qi, F., Guo, S., Guo, B.-N.: Complete monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 233, 2149–2160 (2010)

22. Salem, A.: A completely monotonic function involving q-gamma and q-digamma functions. J. Approx. Theory 164, 971–980 (2012)

23. Sevli, H., Batir, N.: Complete monotonicity results for some functions involving the gamma and polygamma functions. Math. Comput. Model. 53, 1771–1775 (2011)

24. Srivastava, H.M., Guo, S., Qi, F.: Some properties of a class of functions related to completely monotonic functions. Comput. Math. Appl. 64, 1649–1654 (2012)

25. Widder, D.V.: Necessary and sufficient conditions for the representation of a function as a Laplace integral. Trans. Am. Math. Soc. 33, 851–892 (1931)

### Acknowledgements

The authors thank the editor and the reviewers for their valuable suggestions and comments which have improved the manuscript significantly.

Not applicable.

## Funding

The present investigation was supported, in part, by the Natural Science Foundation of China under Grant 11401604.

## Author information

Authors

### 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.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

Dedicated to Professor Hari M. Srivastava on the occasion of his eightieth birthday.

## Rights and permissions

Reprints and permissions

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