![]() Satheesh, S, Sandhya, E: Distributions with completely monotone probability sequences. Sandhya, E, Satheesh, S: On distribution functions with completely monotone derivative. Kuk, AYC: A litter-based approach to risk assessment in developmentaltoxicity studies via a power family of completely monotone functions. Kimberling, CH: Exchangeable events and completely monotonic sequences. Kimberling, CH: A probabilistic interpretation of complete monotonicity. 6,171-173 (1963)Ĥ.ğeller, W: An Introduction to Probability Theory and Its Applications, vol. Lorch, L, Moser, L: A remark on completely monotonic sequences, with an application to summability. Wimp, J: Sequence Transformations and Their Applications. Princeton University Press, Princeton (1946)Ģ. ![]() Received: 6 June 2013 Accepted: 3 September 2013 Published: ġ. The present investigation was supported, in part, by the NaturalScience Foundation of Henan Province ofChina underGrant 112300410022. The authors thank the editor and the referees, one of whom brought our attention to the reference, for their valuable suggestions to improve the quality of this paper. 3Department of Mathematics, Faculty of Arts and Sciences, Nevjehir University, Nevjehir, 50300, Turkey. 2 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada. Author detailsġ Department of Mathematics, Zhongyuan University ofTechnology, Zhengzhou, Henan 450007, People's Republic of China. They also read and approved the finalmanuscript. Authors' contributionsĪll the authors contributed to the writing of the present article. The authors declare that they have no competing interests. By Theorem 11, we know that the condition is sufficient. Proof of Theorem 12 By Definition 2 and by setting m = 0 in Theorem 9, we see that the condition is necessary. Is a convergent positive series, we know thatįrom (22) and (24), we obtain that (-1)kAk^o ^ 0, k e No. ![]() Therefore, by mathematical induction, (22) is valid for all k e N0. Which means that (22) is valid for k = m + 1. Proof of Theorem 11 By the definition of completely monotonic sequence, we only need to prove thatįrom the condition of Theorem 11, (22) is valid for k = 0. Therefore, by mathematical induction, (12) is valid for all k e N0. Which means that (12) is valid for k = r + 1. Which means that (12) is valid for k = 0. Proof of Theorem 10 Let m be a fixed non-negative integer. ![]() Proof of Corollary 1 This corollary can be obtained from (15). The proof of Theorem 9 is thus completed. We first recall some definitions and basic results on or related to completely monotonic sequences and completely monotonic functions.ĭefinition 1 A sequence TO=0 is completely monotonic, by Theorem 2, there exists a non-decreasing and bounded function a(t) on the interval such thatįrom (3), (4) and (13), we can prove that Keywords: necessary condition sufficient condition necessary and sufficient condition difference equation moment sequence completely monotonic sequence completely monotonic function bounded variation Stieltjes integral In this article, we present some necessary conditions, a sufficient condition and a necessary and sufficient condition for sequences to be completely monotonic. "Correspondence: 1 Department of Mathematics, Zhongyuan University of Technology, Zhengzhou, Henan 450007, People's Republic of China Full list of author information is available at the end of the article Senlin Guo1*, Hari M Srivastava2 and Necdet Batir3 To better understanding, we got two sequences for you.A certain class of completely monotonic sequences ![]()
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