 Research
 Open access
 Published:
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.
References
Alzer, H., Batir, N.: Monotonicity properties of the gamma function. Appl. Math. Lett. 20, 778–781 (2007)
Alzer, H., Berg, C., Koumandos, S.: On a conjecture of Clark and Ismail. J. Approx. Theory 134, 102–113 (2005)
Batir, N.: On some properties of the gamma function. Expo. Math. 26, 187–196 (2008)
Bernstein, S.: Sur la définition et les propriétés des fonctions analytiques d’une variable réelle. Math. Ann. 75, 449–468 (1914)
Guo, S.: Logarithmically completely monotonic functions and applications. Appl. Math. Comput. 221, 169–176 (2013)
Guo, S.: Some properties of completely monotonic sequences and related interpolation. Appl. Math. Comput. 219, 4958–4962 (2013)
Guo, S.: A class of logarithmically completely monotonic functions and their applications. J. Appl. Math. 2014, 757462 (2014)
Guo, S.: Some conditions for a class of functions to be completely monotonic. J. Inequal. Appl. 2015, 11 (2015)
Guo, S.: On completely monotonic and related functions. Filomat 30, 2083–2090 (2016)
Guo, S.: Some properties of functions related to completely monotonic functions. Filomat 31, 247–254 (2017)
Guo, S., Laforgia, A., Batir, N., Luo, Q.M.: Completely monotonic and related functions: their applications. J. Appl. Math. 2014, 768516 (2014)
Guo, S., Qi, F.: A class of logarithmically completely monotonic functions associated with the gamma function. J. Comput. Appl. Math. 224, 127–132 (2009)
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)
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)
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)
Guo, S., Srivastava, H.M.: A class of logarithmically completely monotonic functions. Appl. Math. Lett. 21, 1134–1141 (2008)
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)
Guo, S., Srivastava, H.M., Batir, N.: A certain class of completely monotonic sequences. Adv. Differ. Equ. 2013, 294 (2013)
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)
Hausdorff, F.: Summationsmethoden und momentfolgen I. Math. Z. 9, 74–109 (1921)
Qi, F., Guo, S., Guo, B.N.: Complete monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 233, 2149–2160 (2010)
Salem, A.: A completely monotonic function involving qgamma and qdigamma functions. J. Approx. Theory 164, 971–980 (2012)
Sevli, H., Batir, N.: Complete monotonicity results for some functions involving the gamma and polygamma functions. Math. Comput. Model. 53, 1771–1775 (2011)
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)
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.
Availability of data and materials
Not applicable.
Funding
The present investigation was supported, in part, by the Natural Science Foundation of China under Grant 11401604.
Author information
Authors and Affiliations
Contributions
All the authors contributed to the writing of the present article. They also read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Dedicated to Professor Hari M. Srivastava on the occasion of his eightieth birthday.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662020030518