Open Access

A certain class of completely monotonic sequences

Advances in Difference Equations20132013:294

https://doi.org/10.1186/1687-1847-2013-294

Received: 6 June 2013

Accepted: 3 September 2013

Published: 7 November 2013

Abstract

In this article, we present some necessary conditions, a sufficient condition and a necessary and sufficient condition for sequences to be completely monotonic. One counterexample is also presented.

MSC:40A05, 26A45, 26A48, 39A60.

Keywords

necessary conditionsufficient conditionnecessary and sufficient conditiondifference equationmoment sequencecompletely monotonic sequencecompletely monotonic functionbounded variationStieltjes integral

1 Introduction and the main results

We first recall some definitions and basic results on or related to completely monotonic sequences and completely monotonic functions.

Definition 1 [1]

A sequence { μ n } n = 0 is called a moment sequence if there exists a function α ( t ) of bounded variation on the interval [ 0 , 1 ] such that
μ n = 0 1 t n d α ( t ) , n N 0 .
(1)
Here, in Definition 1 and throughout the paper,
N 0 : = { 0 } N ,

and is the set of all positive integers.

Definition 2 [1]

A sequence { μ n } n = 0 is called completely monotonic if
( 1 ) k Δ k μ n 0 , n , k N 0 ,
(2)
where
Δ 0 μ n = μ n
(3)
and
Δ k + 1 μ n = Δ k μ n + 1 Δ k μ n .
(4)

Such a sequence is called totally monotone in [2].

From Definition 2, using mathematical induction, we can prove, for a completely monotonic sequence { μ n } n = 0 , that the sequence { ( 1 ) m Δ m μ n } n = 0 is non-increasing for any fixed m N 0 , and that the sequence { ( 1 ) m Δ m μ n } m = 0 is non-increasing for any fixed n N 0 . The difference equation (4) plays an important role in the proofs of these properties and our main results of this paper.

In [3], the authors showed that for a completely monotonic sequence { μ n } n = 0 , we always have
( 1 ) k Δ k μ n > 0 , n , k N 0 ,
(5)

unless μ n = c , a constant for all n N .

Let
λ k , m : = ( k m ) ( 1 ) k m Δ k m μ m , k , m N 0 .
(6)

It was shown (see [1]) as follows.

Theorem 1 A sequence { μ n } n = 0 is a moment sequence if and only if there exists a constant L such that
m = 0 k | λ k , m | < L , k N 0 ,
(7)

where in (7), λ k , m is defined by (6).

For completely monotonic sequences, the following is the well-known Hausdorff’s theorem (see [1]).

Theorem 2 A sequence { μ n } n = 0 is completely monotonic if and only if there exists a non-decreasing and bounded function α ( t ) on [ 0 , 1 ] such that
μ n = 0 1 t n d α ( t ) , n N 0 .
(8)

From this theorem, we know (see [1]) that a completely monotonic sequence is a moment sequence and is as follows.

Theorem 3 A necessary and sufficient condition that the sequence { μ n } n = 0 should be a moment sequence is that it should be the difference of two completely monotonic sequences.

We also recall the following definition.

Definition 3 [1]

A function f is said to be completely monotonic on an interval I if f is continuous on I has derivatives of all orders on I o (the interior of I) and for all n N 0 ,
( 1 ) n f ( n ) ( x ) 0 , x I o .
(9)

Some mathematicians use the terminology completely monotone instead of completely monotonic. The class of all completely monotonic functions on the interval I is denoted by CM ( I ) .

The completely monotonic functions and completely monotonic sequences have remarkable applications in probability and statistics [410], physics [11, 12], numerical and asymptotic analysis [2], etc.

For the completely monotonic functions on the interval [ 0 , ) , Widder proved (see [1]).

Theorem 4 A function f on the interval [ 0 , ) is completely monotonic if and only if there exists a bounded and non-decreasing function α ( t ) on [ 0 , ) such that
f ( x ) = 0 e x t d α ( t ) .
(10)

There is rich literature on completely monotonic functions. For more recent works, see, for example, [1326].

There exists a close relationship between completely monotonic functions and completely monotonic sequences. For example, Widder [27] showed the following.

Theorem 5 Suppose that f CM [ a , ) , then for any δ 0 , the sequence { f ( a + n δ ) } n = 0 is completely monotonic.

This result was generalized in [28] as follows.

Theorem 6 Suppose that f CM [ a , ) . If the sequence { Δ x k } k = 0 is completely monotonic and x 0 a , then the sequence { f ( x k ) } k = 0 is also completely monotonic.

For the meaning of Δ x k , k N 0 in Theorem 6, see (3) and (4).

Suppose that f CM [ 0 , ) . By Theorem 5, we know that { f ( n ) } n = 0 is completely monotonic.

The following result was obtained in [16].

Theorem 7 Suppose that the sequence { μ n } n = 0 is completely monotonic, then for any ε ( 0 , 1 ) , there exists a continuous interpolating function f ( x ) on the interval [ 0 , ) such that f | [ 0 , ε ] and f | [ ε , ) are both completely monotonic and
f ( n ) = μ n , n N 0 .

From this result or Theorem 2, we can get the following.

Theorem 8 Suppose that the sequence { μ n } n = 0 is completely monotonic. Then there exists a completely monotonic interpolating function g ( x ) on the interval [ 1 , ) such that
g ( n ) = μ n , n N .
It should be noted that (see [[1], Chapter IV]) under the condition of Theorem 8, we cannot guarantee that there exists a completely monotonic interpolating function g ( x ) on the interval [ 0 , ) such that
g ( n ) = μ n , n N 0 .

In this article, we shall further investigate the properties of the completely monotonic sequences. We shall give some necessary conditions, a sufficient condition and a necessary and sufficient condition for sequences to be completely monotonic. More precisely we have the following results.

Theorem 9 Suppose that the sequence { μ n } n = 0 is completely monotonic. Then, for any m N 0 , the series
j = 0 ( 1 ) j Δ j μ m + 1
converges and
μ m j = 0 ( 1 ) j Δ j μ m + 1 .
Corollary 1 Suppose that the sequence { μ n } n = 0 is completely monotonic. Then for m , k N 0 ,
μ m = ( 1 ) k + 1 Δ k + 1 μ m + i = 0 k ( 1 ) i Δ i μ m + 1 .
(11)
Remark 1 Although from the complete monotonicity of the sequence { μ n } n = 0 , we can deduce that for any m N 0 , the series
j = 0 ( 1 ) j Δ j μ m + 1
converges, it cannot guarantee the convergence of the series
j = 0 ( 1 ) j Δ j μ 0 .
For example, let
μ n = 1 n + 1 , n N 0 .
Since the function
f ( x ) = 1 x + 1
is completely monotonic on the interval [ 0 , ) , by Theorem 5, we see that the sequence
{ μ n } n = 0 : = { f ( n ) } n = 0 = { 1 n + 1 } n = 0
is completely monotonic. This conclusion can also be obtained by setting
α ( t ) = t

in Theorem 2.

We can verify that
Δ j μ 0 = ( 1 ) j j + 1 .
Hence,
j = 0 ( 1 ) j Δ j μ 0 = j = 0 1 j + 1

is the famous harmonic series, which is divergent.

Theorem 10 Suppose that the sequence { μ n } n = 0 is completely monotonic. Then for any k , m N 0 ,
( 1 ) k Δ k μ m j = k ( 1 ) j Δ j μ m + 1 .
(12)
Theorem 11 Suppose that the sequence { μ n } n = 1 is completely monotonic and that the series
j = 0 ( 1 ) j Δ j μ 1
converges. Let μ 0 be such that
μ 0 j = 0 ( 1 ) j Δ j μ 1 .

Then the sequence { μ n } n = 0 is completely monotonic.

Theorem 12 A necessary and sufficient condition for the sequence { μ n } n = 0 to be completely monotonic is that the sequence { μ n } n = 1 is completely monotonic, the series
j = 0 ( 1 ) j Δ j μ 1
converges and
μ 0 j = 0 ( 1 ) j Δ j μ 1 .

2 Proofs of the main results

Now, we are in a position to prove the main results.

Proof of Theorem 9 Since { μ n } n = 0 is completely monotonic, by Theorem 2, there exists a non-decreasing and bounded function α ( t ) on the interval [ 0 , 1 ] such that
μ n = 0 1 t n d α ( t ) , n N 0 .
(13)
From (3), (4) and (13), we can prove that
( 1 ) i Δ i μ n = 0 1 ( 1 t ) i t n d α ( t ) , i , n N 0 .
(14)
Now, for k N , we have
i = 0 k 1 ( 1 ) i Δ i μ m + 1 = i = 0 k 1 0 1 ( 1 t ) i t m + 1 d α ( t ) = 0 1 t m + 1 i = 0 k 1 ( 1 t ) i d α ( t ) = 0 1 t m ( 1 ( 1 t ) k ) d α ( t ) = 0 1 t m d α ( t ) 0 1 ( 1 t ) k t m d α ( t ) = μ m ( 1 ) k Δ k μ m , m N 0 .
Hence, for k N ,
μ m = ( 1 ) k Δ k μ m + i = 0 k 1 ( 1 ) i Δ i μ m + 1 , m N 0 .
(15)
Since
( 1 ) i Δ i μ n 0 , i , n N 0 ,
(16)
from (15), we get, for k 1 ,
μ m i = 0 k 1 ( 1 ) i Δ i μ m + 1 , m N 0 .
(17)
From (16), we also know that
j = 0 ( 1 ) j Δ j μ m + 1 , m N 0
is a positive series. Then by (17), we obtain that
j = 0 ( 1 ) j Δ j μ m + 1 , m N 0
converges and that
μ m j = 0 ( 1 ) j Δ j μ m + 1 , m N 0 .
(18)

The proof of Theorem 9 is thus completed. □

Proof of Corollary 1 This corollary can be obtained from (15). □

Proof of Theorem 10 Let m be a fixed non-negative integer.

From Theorem 9, we see that
μ m j = 0 ( 1 ) j Δ j μ m + 1 ,
(19)

which means that (12) is valid for k = 0 .

Suppose that (12) is valid for k = r . Then
( 1 ) r + 1 Δ r + 1 μ m = ( 1 ) r + 1 ( Δ r μ m + 1 Δ r μ m ) = ( 1 ) r ( Δ r μ m Δ r μ m + 1 ) = ( 1 ) r Δ r μ m ( 1 ) r Δ r μ m + 1 j = r ( 1 ) j Δ j μ m + 1 ( 1 ) r Δ r μ m + 1 = j = r + 1 ( 1 ) j Δ j μ m + 1 ,
(20)

which means that (12) is valid for k = r + 1 . Therefore, by mathematical induction, (12) is valid for all k N 0 . The proof of Theorem 10 is completed. □

Proof of Theorem 11 By the definition of completely monotonic sequence, we only need to prove that
( 1 ) k Δ k μ 0 0 , k N 0 .
(21)
We first prove that
( 1 ) k Δ k μ 0 j = k ( 1 ) j Δ j μ 1 , k N 0 .
(22)

From the condition of Theorem 11, (22) is valid for k = 0 .

Suppose that (22) is valid for k = m . Then we have
( 1 ) m + 1 Δ m + 1 μ 0 = ( 1 ) m + 1 ( Δ m μ 1 Δ m μ 0 ) = ( 1 ) m ( Δ m μ 0 Δ m μ 1 ) = ( 1 ) m Δ m μ 0 ( 1 ) m Δ m μ 1 j = m ( 1 ) j Δ j μ 1 ( 1 ) m Δ m μ 1 = j = m + 1 ( 1 ) j Δ j μ 1 ,
(23)

which means that (22) is valid for k = m + 1 . Therefore, by mathematical induction, (22) is valid for all k N 0 .

Since
j = 0 ( 1 ) j Δ j μ 1
is a convergent positive series, we know that
j = k ( 1 ) j Δ j μ 1 0 , k N 0 .
(24)
From (22) and (24), we obtain that
( 1 ) k Δ k μ 0 0 , k N 0 .

The proof of Theorem 11 is completed. □

Proof of Theorem 12 By Definition 2 and by setting m = 0 in Theorem 9, we see that the condition is necessary. By Theorem 11, we know that the condition is sufficient. The proof of Theorem 12 is completed. □

Declarations

Acknowledgements

The authors thank the editor and the referees, one of whom brought our attention to the reference [6], for their valuable suggestions to improve the quality of this paper. The present investigation was supported, in part, by the Natural Science Foundation of Henan Province of China under Grant 112300410022.

Authors’ Affiliations

(1)
Department of Mathematics, Zhongyuan University of Technology
(2)
Department of Mathematics and Statistics, University of Victoria
(3)
Department of Mathematics, Faculty of Arts and Sciences, Nevşehir University

References

  1. Widder DV: The Laplace Transform. Princeton University Press, Princeton; 1946.Google Scholar
  2. Wimp J: Sequence Transformations and Their Applications. Academic Press, New York; 1981.Google Scholar
  3. Lorch L, Moser L: A remark on completely monotonic sequences, with an application to summability. Can. Math. Bull. 1963, 6: 171-173. 10.4153/CMB-1963-016-3MathSciNetView ArticleGoogle Scholar
  4. Feller W 2. In An Introduction to Probability Theory and Its Applications. Wiley, New York; 1966.Google Scholar
  5. Kimberling CH: A probabilistic interpretation of complete monotonicity. Aequ. Math. 1974, 10: 152-164. 10.1007/BF01832852MathSciNetView ArticleGoogle Scholar
  6. Kimberling CH: Exchangeable events and completely monotonic sequences. Rocky Mt. J. Math. 1973, 3: 565-574. 10.1216/RMJ-1973-3-4-565MathSciNetView ArticleGoogle Scholar
  7. Kuk AYC: A litter-based approach to risk assessment in developmental toxicity studies via a power family of completely monotone functions. J. R. Stat. Soc. C 2004, 53: 369-386. 10.1046/j.1467-9876.2003.05369.xMathSciNetView ArticleGoogle Scholar
  8. Sandhya E, Satheesh S: On distribution functions with completely monotone derivative. Stat. Probab. Lett. 1996, 27: 127-129. 10.1016/0167-7152(95)00053-4MathSciNetView ArticleGoogle Scholar
  9. Satheesh S, Sandhya E: Distributions with completely monotone probability sequences. Far East J. Theor. Stat. 1997, 1: 69-75.MathSciNetGoogle Scholar
  10. Trimble SY, Wells J, Wright FT: Superadditive functions and a statistical application. SIAM J. Math. Anal. 1989, 20: 1255-1259. 10.1137/0520082MathSciNetView ArticleGoogle Scholar
  11. Day WA: On monotonicity of the relaxation functions of viscoelastic materials. Proc. Camb. Philos. Soc. 1970, 67: 503-508. 10.1017/S0305004100045771View ArticleGoogle Scholar
  12. Franosch T, Voigtmann T: Completely monotone solutions of the mode-coupling theory for mixtures. J. Stat. Phys. 2002, 109: 237-259. 10.1023/A:1019991729106MathSciNetView ArticleGoogle Scholar
  13. Alzer H, Batir N: Monotonicity properties of the gamma function. Appl. Math. Lett. 2007, 20: 778-781. 10.1016/j.aml.2006.08.026MathSciNetView ArticleGoogle Scholar
  14. Batir N: On some properties of the gamma function. Expo. Math. 2008, 26: 187-196. 10.1016/j.exmath.2007.10.001MathSciNetView ArticleGoogle Scholar
  15. Guo S: Logarithmically completely monotonic functions and applications. Appl. Math. Comput. 2013, 221: 169-176.MathSciNetView ArticleGoogle Scholar
  16. Guo S: Some properties of completely monotonic sequences and related interpolation. Appl. Math. Comput. 2013, 219: 4958-4962. 10.1016/j.amc.2012.11.073MathSciNetView ArticleGoogle Scholar
  17. Guo S, Qi F: A class of logarithmically completely monotonic functions associated with the gamma function. J. Comput. Appl. Math. 2009, 224: 127-132. 10.1016/j.cam.2008.04.028MathSciNetView ArticleGoogle Scholar
  18. Guo S, Qi F, Srivastava HM: A class of logarithmically completely monotonic functions related to the gamma function with applications. Integral Transforms Spec. Funct. 2012, 23: 557-566. 10.1080/10652469.2011.611331MathSciNetView ArticleGoogle Scholar
  19. Guo S, Qi F, Srivastava HM: Supplements to a class of logarithmically completely monotonic functions associated with the gamma function. Appl. Math. Comput. 2008, 197: 768-774. 10.1016/j.amc.2007.08.011MathSciNetView ArticleGoogle Scholar
  20. Guo S, Qi F, Srivastava HM: Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic. Integral Transforms Spec. Funct. 2007, 18: 819-826. 10.1080/10652460701528933MathSciNetView ArticleGoogle Scholar
  21. Guo S, Srivastava HM: A certain function class related to the class of logarithmically completely monotonic functions. Math. Comput. Model. 2009, 49: 2073-2079. 10.1016/j.mcm.2009.01.002MathSciNetView ArticleGoogle Scholar
  22. Guo S, Srivastava HM: A class of logarithmically completely monotonic functions. Appl. Math. Lett. 2008, 21: 1134-1141. 10.1016/j.aml.2007.10.028MathSciNetView ArticleGoogle Scholar
  23. Guo S, Xu J-G, Qi F: Some exact constants for the approximation of the quantity in the Wallis’ formula. J. Inequal. Appl. 2013., 2013: Article ID 67Google Scholar
  24. Qi F, Guo S, Guo B-N: Complete monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 2010, 233: 2149-2160. 10.1016/j.cam.2009.09.044MathSciNetView ArticleGoogle Scholar
  25. Sevli H, Batir N: Complete monotonicity results for some functions involving the gamma and polygamma functions. Math. Comput. Model. 2011, 53: 1771-1775. 10.1016/j.mcm.2010.12.055MathSciNetView ArticleGoogle Scholar
  26. Srivastava HM, Guo S, Qi F: Some properties of a class of functions related to completely monotonic functions. Comput. Math. Appl. 2012, 64: 1649-1654. 10.1016/j.camwa.2012.01.016MathSciNetView ArticleGoogle Scholar
  27. Widder DV: Necessary and sufficient conditions for the representation of a function as a Laplace integral. Trans. Am. Math. Soc. 1931, 33: 851-892. 10.1090/S0002-9947-1931-1501621-6MathSciNetView ArticleGoogle Scholar
  28. Lorch L, Newman DJ: On the composition of completely monotonic functions, completely monotonic sequences and related questions. J. Lond. Math. Soc. 1983, 28: 31-45.MathSciNetView ArticleGoogle Scholar

Copyright

© Guo et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.