- Research
- Open Access
Some conditions for a class of functions to be completely monotonic
- Senlin Guo^{1}Email author
https://doi.org/10.1186/s13660-014-0534-y
© Guo; licensee Springer 2015
- Received: 3 November 2014
- Accepted: 17 December 2014
- Published: 13 January 2015
Abstract
In this article, we present a necessary condition and a necessary and sufficient condition for a class of functions to be completely monotonic.
Keywords
- necessary condition
- necessary and sufficient condition
- completely monotonic function
- gamma function
MSC
- 34A40
- 26D10
- 26A48
1 Introduction and main results
It was proved [4] that a logarithmically completely monotonic function is also completely monotonic.
There is a rich literature on completely monotonic, logarithmically completely monotonic functions and their applications. For more recent work, see, for example, [5–28].
Theorem 1
Corollary 1
Theorem 2
2 Lemmas
We need the following lemmas to prove our main results.
Lemma 1
(see [11])
Lemma 2
(see [7])
3 Proof of the main results
Proof of Theorem 1
The proof of Theorem 1 is thus completed. □
Proof of Theorem 2
By Theorem 1, the condition is necessary.
The proof of Theorem 2 is hence completed. □
Declarations
Acknowledgements
The author thanks the editor and the referees for their valuable suggestions to improve the quality of this paper.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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