Journal of Applied Mathematics

Volume 2014, Article ID 768516, 2 pages

http://dx.doi.org/10.1155/2014/768516

## Completely Monotonic and Related Functions: Their Applications

^{1}Department of Mathematics, Zhongyuan University of Technology, Zhengzhou, Henan 450007, China^{2}Department of Mathematics, Roma Tre University, Largo San Leonardo Murialdo 1, 00146 Rome, Italy^{3}Department of Mathematics, Faculty of Arts and Sciences, Nevsehir University, 50300 Nevsehir, Turkey^{4}Department of Mathematics, Chongqing Normal University, Chongqing 401331, China

Received 1 October 2014; Accepted 1 October 2014; Published 31 December 2014

Copyright © 2014 Senlin Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Completely monotonic and related functions are important function classes in mathematical analysis. It was Bernstein [1] who in 1914 first introduced the notion of* completely monotonic function*. This year we celebrate its 100th anniversary. In 1921, Hausdorff [2] gave the notion of* completely monotonic sequence*, which is related to the notion of completely monotonic function. Widder [3] in 1931 introduced the notion of* minimal completely monotonic sequence*. The terminology:* logarithmically completely monotonic function*, was first used in [4] in 1988. In 1989, the authors [5] introduced the notion of* strongly completely monotonic function*. In 2009, the authors [6] introduced the notion of* strongly logarithmically completely monotonic function*. Also, the terminology:* almost strongly completely monotonic function*, was introduced in [6] in order to simplify the statement of the main results of the article [6]. In 2012, the terminology of* almost completely monotonic function* was defined and used in [7] in order to simplify the statement of the main results of the article [7]. Researchers did a lot of investigations, in both theory and applications, on these functions. More recently, for example, the authors [6] showed that a strongly logarithmically completely monotonic function must be almost strongly completely monotonic. Also, in [6], the following fact that a strongly logarithmically completely monotonic function cannot be strongly completely monotonic, or, equivalently, a strongly completely monotonic function cannot be strongly logarithmically completely monotonic was established. Figure 1 illustrates the inclusion relationships among these classes of functions on . In these inclusion relationships, the fact that a logarithmically completely monotonic function must be completely monotonic was, in fact, first proved in [8] although the author did not use the terminology of* logarithmically completely monotonic function*; all others are from [6] or directly from their definitions. Also note that in [6] counter examples were presented to show that some function classes are not included in some other function classes or to show that the intersection sets of two function classes are not empty.