research article complete monotonicity of functions

Research ArticleComplete Monotonicity of Functions Connected withthe Exponential Function and Derivatives

Chun-Fu Wei12 and Bai-Ni Guo2

1 State Key Laboratory Cultivation Base for Gas Geology and Gas Control Henan Polytechnic UniversityJiaozuo Henan 454010 China

2 School of Mathematics and Informatics Henan Polytechnic University JiaozuoHenan 454010 China

Correspondence should be addressed to Chun-Fu Wei mathwcf163com

Received 25 January 2014 Accepted 20 March 2014 Published 15 April 2014

Academic Editor Beong In Yun

Copyright copy 2014 C-F Wei and B-N Guo This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Some completemonotonicity results that the functionsplusmn1 (119890plusmn119905 minus 1) are logarithmically completelymonotonic and that differencesbetween consecutive derivatives of these two functions are completely monotonic and that the ratios between consecutivederivatives of these two functions are decreasing on (0infin) are discovered As applications of these newly discovered results somecomplete monotonicity results concerning the polylogarithm are found Finally a conjecture on the complete monotonicity of theabove-mentioned ratios is posed

1 Introduction and Main Results

Throughout this paper we denote the set of all positive inte-gers by N

Recently the following problem was posed in [1 page569] For 119905 = 0 and 119896 isin N determine the numbers 119886

119896119894minus1for

1 le 119894 le 119896 such that

1

(1 minus 119890minus119905)119896= 1 +

119896

sum

119894=1

119886119896119894minus1

(1

119890119905minus 1

)

(119894minus1)

(1)

This problem among other things was answered in [1] byeight identities Two of the eight identities may be recited asfollows

Theorem A (see [1 Theorem 21]) For 119894 isin 0 cup N one has

(1

119890119905minus 1

)

(119894)

=

119894+1

sum

119898=1

120582119894119898

(119890119905minus 1)119898 (2)

where

120582119894119898

= (minus1)119894(119898 minus 1)119878 (119894 + 1119898)

119878 (119894 119898) =1

119898

119898

sum

ℓ=1

(minus1)119898minusℓ

(119898

ℓ) ℓ119894 1 le 119898 le 119894

(3)

are Stirling numbers of the second kind

TheoremB (see [1Theorem 23]) For 119894 119896 isin Nwith 1 le 119894 le 119896the coefficients 119886

119896119894minus1defined in (1)may be calculated by

119886119896119894minus1

= (minus1)1198942+1119872119896minus119894+1

(119896 119894) (4)

where

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 851213 5 pageshttpdxdoiorg1011552014851213

2 Abstract and Applied Analysis

119872119895(119896 119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

(119894 minus 1)

(119896

119894) 119878 (119894 + 1 119894) sdot sdot sdot 119878 (119894 + 119895 minus 1 119894)

1

119894

(119896

119894 + 1) 119878 (119894 + 1 119894 + 1) sdot sdot sdot 119878 (119894 + 119895 minus 1 119894 + 1)

1

(119894 + 1)

(119896

119894 + 2) 0 sdot sdot sdot 119878 (119894 + 119895 minus 1 119894 + 2)

d

1

(119894 + 119895 minus 2)

(119896

119894 + 119895 minus 1) 0 sdot sdot sdot 119878 (119894 + 119895 minus 1 119894 + 119895 minus 1)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(5)

For more information please refer to [2 3] and relatedreferences therein

Stimulated by results obtained in [1] as mentioned abovewe are interested in two functional sequences

119865119894(119905) = (minus1)

119894(

1

119890119905minus 1

)

(119894)

119866119894(119905) = (minus1)

119894(

1

1 minus 119890minus119905)

(119894)

(6)

where 119905 isin (0infin) and 119894 isin 0 cup N and we firstly discover inthis paper the following results

Theorem 1 For 119894 isin 0 cup N the functions 119865119894(119905) and 119866

119894(119905) are

completely monotonic on (0infin) More strongly the functions1198650(119905) and 119866

0(119905) are logarithmically completely monotonic on

(0infin)For 119894 119896 isin 0 cup N the functions 119865

119894(119905) and 119866

119894(119905) satisfy

119865(119896)

119894(119905) = (minus1)

119896119865119894+119896

(119905)

119866(119896)

119894(119905) = (minus1)

119896119866119894+119896

(119905)

(7)

Theorem 2 For given 119894 isin 0 cup N the differences

F119894(119905) = 119865

119894+1(119905) minus 119865

119894(119905)

G0(119905) = 119866

1(119905) minus 119866

0(119905) + 1

G119894+1

(119905) = 119866119894+2

(119905) minus 119866119894+1

(119905)

(8)

are completely monotonic functions on (0infin) In particularthe sequences 119865

119894(119905) and119866

119894(119905) are increasing with respect to 119894 for

119905 isin (0infin) that is the inequalities

119865119894(119905) lt 119865

119894+1(119905) 119866

119894(119905) lt 119866

119894+1(119905) (9)

are valid for all 119894 isin 0 cup N and 119905 isin (0infin)

Theorem 3 For given 119894 isin 0 cup N the ratios

F119894(119905) =

119865119894+1

(119905)

119865119894(119905)

G119894(119905) =

119866119894+1

(119905)

119866119894(119905)

(10)

are decreasing on (0infin) with

lim119905rarrinfin

F119894(119905) = 1 lim

119905rarrinfin

G0(119905) = 0 lim

119905rarrinfin

G119894+1

(119905) = 1

(11)

In order to further show the importance and significanceof the two functional sequences in (6) we secondly give sev-eral applications of Theorems 1 to 3 in Section 4

Finally we pose a conjecture on the completemonotonic-ity of the functionsF

119894(119905) andG

119894(119905) defined in (10)

2 Two Definitions and a Lemma

Now we list definitions of the completely monotonic and thelogarithmically completely monotonic functions which justnow appeared inTheorems 1 and 2 and recite a lemma whichis needed to proveTheorem 3

Definition 4 (see [4 5]) A function 119902(119909) is said to be com-pletely monotonic on an interval 119868 if 119902(119909) has derivatives ofall orders on 119868 and (minus1)119899119902(119899)(119909) ge 0 for 119909 isin 119868 and 119899 ge 0

The noted Hausdorff-Bernstein-Widder theorem [5 page161 Theorem 12b] says that a necessary and sufficient condi-tion that 119891(119909) should be completely monotonic for 0 lt 119909 lt

infin is that

119891 (119909) = int

infin

0

119890minus119909119905d120572 (119905) (12)

where 120572(119905) is nondecreasing and the integral converges for0 lt 119909 lt infin In other words a function 119891 defined on (0infin) iscompletely monotonic on (0infin) if and only if it is a Laplacetransform Formore information on the theory of completelymonotonic functions please refer to [4 Chapter XIII] [5Chapter IV] and the newly published monograph [6] Thismeans that it is useful to confirm the complete monotonicityof functions

Definition 5 (see [7 8]) A positive function119891(119909) is said to belogarithmically completely monotonic on an interval 119868 sube R

if it has derivatives of all orders on 119868 and its logarithm ln119891(119909)satisfies (minus1)119896[ln119891(119909)](119896) ge 0 for 119896 isin N on 119868

It has been proved that any logarithmically completelymonotonic function on 119868 is also completely monotonic on 119868

Abstract and Applied Analysis 3

but not conversely For more information on this class offunctions please refer to [7ndash9] and [10 Section 13] andclosely related references thereinThis shows that it is helpfulto prove the logarithmically complete monotonicity of func-tions

Lemma 6 (see [11 Lemma 22]) Suppose that 119886119896 119887119896gt 0 and

that 119906119896(119905) is a sequence of positive and differentiable functions

such that the seriesinfin

sum

119896=0

119886119896119906(ℓ)

119896(119905)

infin

sum

119896=0

119887119896119906(ℓ)

119896(119905) ℓ = 0 1 (13)

converge absolutely and uniformly over compact subsets of[0infin)

(1) If the logarithmic derivative 1199061015840119896(119905)119906119896(119905) forms an in-

creasing sequence of functions and if 119886119896119887119896decreases

(resp increases) then

119891 (119905) =

suminfin

119896=0119886119896119906119896(119905)

suminfin

119896=0119887119896119906119896(119905)

(14)

decreases (resp increases) for 119905 ge 0(2) If the logarithmic derivative 1199061015840

119896(119905)119906119896(119905) forms a de-


(resp increases) then the function119891(119905) increases (respdecreases) for 119905 ge 0

3 Proofs of Main Results

Now we start out to prove our theorems

Proof of Theorem 1 Since

1198650(119905) = 119866

0(119905) minus 1 =

infin

sum

ℓ=1

119890minusℓ119905

119865119894(119905) = 119866

119894(119905) =

infin

sum

ℓ=1

ℓ119894119890minusℓ119905

(15)

for 119894 ge 1 the functions 119865119894(119905) and 119866

119894(119905) for 119894 isin 0 cup N are

completely monotonic on (0infin)Taking the logarithms of the functions119865

0(119905) and119866

0(119905) and

differentiating yield

[ln1198650(119905)]1015840

=dd119905(ln 1

119890119905minus 1

) = minus119890119905

119890119905minus 1

= minus [1198650(119905) + 1]

[ln1198660(119905)]1015840

=dd119905(ln 1

1 minus 119890minus119905) = minus119865

0(119905)

(16)

Consequently by definition of logarithmically completelymonotonic functions and the above obtained complete mon-otonicity of the function 119865

0(119905) it is ready to deduce the log-

arithmically complete monotonicity of 1198650(119905) and 119866

0(119905) on

(0infin)The formulas in (7) can be straightforwardly verifiedThe

proof of Theorem 1 is complete

Proof of Theorem 2 A simple computation yields

119865119894+1

(119905) minus 119865119894(119905)

= (minus1)119894+1

[(1

119890119905minus 1

)

(119894+1)

+ (1

119890119905minus 1

)

(119894)

]

= (minus1)119894+1 [

[

(

infin

sum

ℓ=1

119890minusℓ119905)

(119894+1)

+ (

infin

sum

ℓ=1

119890minusℓ119905)

(119894)

]

]

= (minus1)119894+1

[

infin

sum

ℓ=1

(minus1)119894+1ℓ119894+1119890minusℓ119905

+

infin

sum

ℓ=1

(minus1)119894ℓ119894119890minusℓ119905]

=

infin

sum

ℓ=2

(ℓ minus 1) ℓ119894119890minusℓ119905

gt 0

(minus1)119899[119865119894+1

(119905) minus 119865119894(119905)](119899)

=

infin

sum

ℓ=2

(ℓ minus 1) ℓ119899+119894119890minusℓ119905

gt 0

(17)

for 119899 isin N By definition the difference F119894(119905) = 119865

119894+1(119905) minus 119865

119894(119905)

for 119894 isin 0 cup N is completely monotonic on (0infin)From the first equalities in (15) respectively it follows that

the functions

G0(119905) = 119866

1(119905) minus 119866

0(119905) + 1 = 119865

1(119905) minus 119865

0(119905)

G119894+1

(119905) = 119866119894+2

(119905) minus G119894+1

(119905) = 119865119894+2

(119905) minus 119865119894+1

(119905)

(18)

are all completely monotonic on (0infin)The inequalities in (9) follow from the positivity of F

119894(119905)

andG119894(119905) for 119894 isin N The proof of Theorem 2 is complete

Proof of Theorem 3 It is easy to see that

F119894(119905) = minus

[1 (119890119905minus 1)](119894+1)

[1 (119890119905minus 1)](119894)

= minus

(suminfin

119896=1119890minus119896119905)

(119894+1)

(suminfin

119896=1119890minus119896119905)(119894)

= minus

suminfin

119896=1(minus1)119894+1119896119894+1119890minus119896119905

suminfin

119896=1(minus1)119894119896119894119890minus119896119905

=

suminfin

119896=1119896119894+1119890minus119896119905

suminfin

119896=1119896119894119890minus119896119905

(19)

Let 119886119896= 119896119894+1 119887119896= 119896119894 and 119906

119896(119905) = 119890

minus119896119905 Then 119886119896119887119896= 119896 is an

increasing sequence and 1199061015840119896(119905)119906119896(119905) = minus119896 form a decreasing

sequence By Lemma 6 it follows that the function F119894(119905) is

decreasing on (0infin)Taking 119905 rarr infin at the very ends of (19) shows that

lim119905rarrinfin

F119894(119905) = lim119905rarrinfin

suminfin

119896=1119896119894+1119890minus119896119905

suminfin

119896=1119896119894119890minus119896119905

= lim119905rarrinfin

1 + suminfin

119896=2119896119894+1119890(1minus119896)119905

1 + suminfin

119896=2119896119894119890(1minus119896)119905

= 1

(20)

The first limit in (11) thus followsMaking use of the second relation in (15) yields G

119894(119905) =

F119894(119905) for 119894 isin N Accordingly the functionG

119894(119905) has the same

monotonicity and the same limit for 119905 rarr infin asF119894(119905) does for


119894 isin N As a result the function G119894(119905) is decreasing on (0infin)

and the third limit in (11) is valid for 119894 isin NA straightforward computation gives

G0(119905) =

1

119890119905minus 1

= 1198650(119905) (21)

which obviously tends to 0 as 119905 rarr infin and apparently de-creases on (0infin) The proof of Theorem 3 is complete

4 Some Applications

We recall from [12] that the polylogarithm Li119904(119911) is the func-

tion

Li119904(119911) =

infin

sum

119896=1

119911119896

119896119904

(22)

defined for all 119904 isin C and over the open unit disk |119911| lt 1 in thecomplex plane C Its definition on the whole complex planethen follows uniquely via analytic continuation

For given 119894 isin 0cupN since 119865119894(119905) = sum

infin

ℓ=1ℓ119894119890minusℓ119905 we observe

that

119865119894(119905) = Li

minus119894(119890minus119905) 119866

119894+1(119905) = Li

minus(119894+1)(119890minus119905) (23)

Therefore considering (15) Theorems 1 to 3 can be used tofind some properties of the polylogarithm Li

119904(119911) as follows

Theorem 7 For 119894 isin 0 cup N the polylogarithm Liminus119894(119890minus119905) is

completelymonotonic with respect to 119905 isin (0infin) More stronglythe polylogarithm Li

0(119890minus119905) is logarithmically completely mono-

tonic with respect to 119905 isin (0infin)For 119894 119896 isin 0 cup N the polylogarithm Li

minus119894(119890minus119905) satisfies

[Liminus119894(119890minus119905)]

(119896)

= (minus1)119896Liminus(119894+119896)

(119890minus119905) (24)


F119894(119905) = Li

minus(119894+1)(119890minus119905) minus Li

minus119894(119890minus119905)

G119894(119905) =

Liminus119894(119890minus119905) minus Li0(119890minus119905) + 1 119894 = 0

F119894+1

(119905) 119894 ge 1

(25)

are completely monotonic functions on (0infin) In particularthe polylogarithm sequence Li

minus119894(119890minus119905) is increasing with respect

to 119894 for given 119905 isin (0infin) that is the inequality

Liminus119894(119890minus119905) lt Li

minus(119894+1)(119890minus119905) (26)

is valid for all 119894 isin 0 cup N and 119905 isin (0infin)


F119894(119905) =

Liminus(119894+1)

(119890minus119905)

Liminus119894(119890minus119905)

G119894(119905) =

F119894(119905) 119894 ge 1

Liminus119894(119890minus119905)

1 + Li0(119890minus119905)

119894 = 0

(27)


lim119905rarrinfin

F119894(119905) = 1 lim

119905rarrinfin

G0(119905) = 0 lim

119905rarrinfin

G119894+1

(119905) = 1

(28)

Furthermore by the above (logarithmically) completemonotonicity in Theorems 7 and 8 and by some completemonotonicity properties of composite functions we can ob-tain the complete monotonicity of functions involving thepolylogarithm Li

minus119894(1119905) for 119894 isin 0 cup N as follows

Theorem 10 The following complete monotonicity is valid

(1) The polylogarithm Li0(1119905) = 1(119905 minus 1) is logarithmi-

cally completely monotonic with respect to 119905 isin (1infin)

(2) For 119894 isin 0 cup N the polylogarithm Liminus119894(1119905) is com-

pletely monotonic with respect to 119905 isin (1infin)

(3) For given 119894 isin 0 cup N the differences

P119894(119905) = Li

minus(119894+1)(1

119905

) minus Liminus119894(1

119905

)

Q119894(119905) =

Liminus119894(1

119905

) minus Li0(1

119905

) + 1 119894 = 0

P119894+1

(119905) 119894 ge 1

(29)

are completely monotonic functions on (1infin)

Proof The second item of [13 Theorem 5] tells us that ifℎ1015840(119909) is completely monotonic on an interval 119868 and 119891(119909)

is logarithmically completely monotonic on the domainℎ(119868) then the composite function 119891 ∘ ℎ(119909) = 119891(ℎ(119909)) islogarithmically completelymonotonic on 119868 It is easy to verifythat the derivative of ln 119905 is 1119905 and completely monotonicon (0infin) Combining these conclusions with the logarithmi-cally complete monotonicity of Li

0(119890minus119905) in Theorem 7 yields

that the polylogarithm Li0(1119905) = 1(119905 minus 1) is logarithmically

completely monotonic with respect to 119905 isin (1infin)In [14 page 83] it was given that if 119891 and 119892 are functions

such that 119891(119892(119909)) is defined on (0infin) and if 119891 and 1198921015840 are

completely monotonic then 119909 997891rarr 119891(119892(119909)) is also completelymonotonic on (0infin) Replacing 119891(119905) by Li

minus119894(119890minus119905) and 119892(119905)

by ln 119905 and making use of the complete monotonicity ofLiminus119894(119890minus119905) in Theorem 7 lead to the complete monotonicity of

the polylogarithm Liminus119894(1119905)

The leftover proofs are similar to the above argumentsThe proof of Theorem 10 is complete

Remark 11 To the best of our knowledge the above (log-arithmically) complete monotonicity results concerning thepolylogarithm are new This shows us the importance of thetwo functional sequences in (6) and the significance ofTheorems 1 to 3


5 A Conjecture

ByTheorem 1 and the fact that any logarithmically completelymonotonic function on 119868 is also completely monotonic on 119868it is clear that the functions

G0(119905) = F

0(119905) minus 1 =

1

119890119905minus 1

= 1198650(119905)

G1(119905) = F

1(119905) = 1 +

2

119890119905minus 1

= 1 + 21198650(119905)

(30)

are all completely monotonic on (0infin) This motivates us topose the following conjecture

Conjecture 12 For given 119894 isin 0 cup N the functions F119894(119905)

and G119894(119905) defined in (10) or (27) are completely monotonic on

(0infin)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for theirvaluable comments on and careful corrections to the originalversion of this paper The authors appreciate Professor DrFeng Qi in China for his valuable contribution to this paperChun-Fu Wei was partially supported by the NNSF underGrant no 51274086 of China by the Ministry of EducationDoctoral Foundation of ChinamdashPriority Areas under Grantno 20124116130001 and by the State Key Laboratory Cultiva-tion Base for Gas Geology and Gas Control under Grant noWS2012A10 at Henan Polytechnic University China

References

[1] B-N Guo and F Qi ldquoSome identities and an explicit formulafor Bernoulli and Stirling numbersrdquo Journal of Computationaland Applied Mathematics vol 255 pp 568ndash579 2014

[2] F Qi ldquoExplicit formulas for computing Euler polynomials interms of the second kind Stirling numbers rdquo httparxivorgabs13105921

[3] A-MXu andZ-DCen ldquoSome identities involving exponentialfunctions and Stirling numbers and applicationsrdquo Journal ofComputational and Applied Mathematics vol 260 pp 201ndash2072014

[4] D SMitrinovic J E Pecaric and AM FinkClassical and NewInequalities in Analysis vol 61 Kluwer Academic Publishers1993

[5] D V Widder The Laplace Transform Princeton UniversityPress Princeton NJ USA 1946

[6] R L Schilling R Song and Z Vondracek Bernstein Functionsvol 37 of de Gruyter Studies inMathematics De Gruyter BerlinGermany 2010

[7] R D Atanassov and U V Tsoukrovski ldquoSome properties ofa class of logarithmically completely monotonic functionsrdquoComptes Rendus de lrsquoAcademie Bulgare des Sciences vol 41 no2 pp 21ndash23 1988

[8] C Berg ldquoIntegral representation of some functions related tothe gamma functionrdquo Mediterranean Journal of Mathematicsvol 1 no 4 pp 433ndash439 2004

[9] F Qi and Q M Luo ldquoBounds for the ratio of two gamma func-tions fromWendelrsquos asymptotic relation to Elezovic-Giordano-Pecaricrsquos theoremrdquo Journal of Inequalities and Applications vol2013 article 542 20 pages 2013

[10] F Qi and Q-M Luo ldquoBounds for the ratio of two gamma func-tions fromWendelrsquos and related inequalities to logarithmicallycompletely monotonic functionsrdquo Banach Journal of Mathe-matical Analysis vol 6 no 2 pp 132ndash158 2012

[11] S Koumandos and H L Pedersen ldquoOn the asymptotic expan-sion of the logarithm of Barnes triple gamma functionrdquoMathe-matica Scandinavica vol 105 no 2 pp 287ndash306 2009

[12] Wikipedia ldquoThe Free Encyclopediardquo httpenwikipediaorgwikiPolylog

[13] F Qi and B-N Guo ldquoSome logarithmically completely mono-tonic functions related to the gamma functionrdquo Journal of theKorean Mathematical Society vol 47 no 6 pp 1283ndash1297 2010

[14] S Bochner Harmonic Analysis and the Theory of ProbabilityCaliforniaMonographs inMathematical Sciences University ofCalifornia Press Berkeley Calif USA 1955

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Stochastic AnalysisInternational Journal of


119872119895(119896 119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

(119894 minus 1)

(119896

119894) 119878 (119894 + 1 119894) sdot sdot sdot 119878 (119894 + 119895 minus 1 119894)

1

119894

(119896

119894 + 1) 119878 (119894 + 1 119894 + 1) sdot sdot sdot 119878 (119894 + 119895 minus 1 119894 + 1)

1

(119894 + 1)

(119896

119894 + 2) 0 sdot sdot sdot 119878 (119894 + 119895 minus 1 119894 + 2)

d

1

(119894 + 119895 minus 2)

(119896

119894 + 119895 minus 1) 0 sdot sdot sdot 119878 (119894 + 119895 minus 1 119894 + 119895 minus 1)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(5)

For more information please refer to [2 3] and relatedreferences therein

Stimulated by results obtained in [1] as mentioned abovewe are interested in two functional sequences

119865119894(119905) = (minus1)

119894(

1

119890119905minus 1

)

(119894)

119866119894(119905) = (minus1)

119894(

1

1 minus 119890minus119905)

(119894)

(6)

where 119905 isin (0infin) and 119894 isin 0 cup N and we firstly discover inthis paper the following results

Theorem 1 For 119894 isin 0 cup N the functions 119865119894(119905) and 119866

119894(119905) are

completely monotonic on (0infin) More strongly the functions1198650(119905) and 119866

0(119905) are logarithmically completely monotonic on

(0infin)For 119894 119896 isin 0 cup N the functions 119865

119894(119905) and 119866

119894(119905) satisfy

119865(119896)

119894(119905) = (minus1)

119896119865119894+119896

(119905)

119866(119896)

119894(119905) = (minus1)

119896119866119894+119896

(119905)

(7)


F119894(119905) = 119865

119894+1(119905) minus 119865

119894(119905)

G0(119905) = 119866

1(119905) minus 119866

0(119905) + 1

G119894+1

(119905) = 119866119894+2

(119905) minus 119866119894+1

(119905)

(8)

are completely monotonic functions on (0infin) In particularthe sequences 119865

119894(119905) and119866

119894(119905) are increasing with respect to 119894 for

119905 isin (0infin) that is the inequalities

119865119894(119905) lt 119865

119894+1(119905) 119866

119894(119905) lt 119866

119894+1(119905) (9)

are valid for all 119894 isin 0 cup N and 119905 isin (0infin)


F119894(119905) =

119865119894+1

(119905)

119865119894(119905)

G119894(119905) =

119866119894+1

(119905)

119866119894(119905)

(10)


lim119905rarrinfin

F119894(119905) = 1 lim

119905rarrinfin

G0(119905) = 0 lim

119905rarrinfin

G119894+1

(119905) = 1

(11)

In order to further show the importance and significanceof the two functional sequences in (6) we secondly give sev-eral applications of Theorems 1 to 3 in Section 4

Finally we pose a conjecture on the completemonotonic-ity of the functionsF

119894(119905) andG

119894(119905) defined in (10)

2 Two Definitions and a Lemma

Now we list definitions of the completely monotonic and thelogarithmically completely monotonic functions which justnow appeared inTheorems 1 and 2 and recite a lemma whichis needed to proveTheorem 3

Definition 4 (see [4 5]) A function 119902(119909) is said to be com-pletely monotonic on an interval 119868 if 119902(119909) has derivatives ofall orders on 119868 and (minus1)119899119902(119899)(119909) ge 0 for 119909 isin 119868 and 119899 ge 0

The noted Hausdorff-Bernstein-Widder theorem [5 page161 Theorem 12b] says that a necessary and sufficient condi-tion that 119891(119909) should be completely monotonic for 0 lt 119909 lt

infin is that

119891 (119909) = int

infin

0

119890minus119909119905d120572 (119905) (12)

where 120572(119905) is nondecreasing and the integral converges for0 lt 119909 lt infin In other words a function 119891 defined on (0infin) iscompletely monotonic on (0infin) if and only if it is a Laplacetransform Formore information on the theory of completelymonotonic functions please refer to [4 Chapter XIII] [5Chapter IV] and the newly published monograph [6] Thismeans that it is useful to confirm the complete monotonicityof functions

Definition 5 (see [7 8]) A positive function119891(119909) is said to belogarithmically completely monotonic on an interval 119868 sube R

if it has derivatives of all orders on 119868 and its logarithm ln119891(119909)satisfies (minus1)119896[ln119891(119909)](119896) ge 0 for 119896 isin N on 119868

It has been proved that any logarithmically completelymonotonic function on 119868 is also completely monotonic on 119868






sum

119896=0

119886119896119906(ℓ)

119896(119905)

infin

sum

119896=0

119887119896119906(ℓ)

119896(119905) ℓ = 0 1 (13)





119891 (119905) =

suminfin

119896=0119886119896119906119896(119905)

suminfin

119896=0119887119896119906119896(119905)

(14)


119896(119905)119906119896(119905) forms a de-






1198650(119905) = 119866

0(119905) minus 1 =

infin

sum

ℓ=1

119890minusℓ119905

119865119894(119905) = 119866

119894(119905) =

infin

sum

ℓ=1

ℓ119894119890minusℓ119905

(15)


119894(119905) for 119894 isin 0 cup N are


0(119905) and119866

0(119905) and


[ln1198650(119905)]1015840

=dd119905(ln 1

119890119905minus 1

) = minus119890119905

119890119905minus 1

= minus [1198650(119905) + 1]

[ln1198660(119905)]1015840

=dd119905(ln 1


0(119905)

(16)




0(119905) on




119865119894+1

(119905) minus 119865119894(119905)

= (minus1)119894+1

[(1

119890119905minus 1

)

(119894+1)

+ (1

119890119905minus 1

)

(119894)

]

= (minus1)119894+1 [

[

(

infin

sum

ℓ=1

119890minusℓ119905)

(119894+1)

+ (

infin

sum

ℓ=1

119890minusℓ119905)

(119894)

]

]

= (minus1)119894+1

[

infin

sum

ℓ=1

(minus1)119894+1ℓ119894+1119890minusℓ119905

+

infin

sum

ℓ=1

(minus1)119894ℓ119894119890minusℓ119905]

=

infin

sum

ℓ=2

(ℓ minus 1) ℓ119894119890minusℓ119905

gt 0

(minus1)119899[119865119894+1

(119905) minus 119865119894(119905)](119899)

=

infin

sum

ℓ=2

(ℓ minus 1) ℓ119899+119894119890minusℓ119905

gt 0

(17)


119894+1(119905) minus 119865

119894(119905)


the functions

G0(119905) = 119866

1(119905) minus 119866

0(119905) + 1 = 119865

1(119905) minus 119865

0(119905)

G119894+1

(119905) = 119866119894+2

(119905) minus G119894+1

(119905) = 119865119894+2

(119905) minus 119865119894+1

(119905)

(18)


119894(119905)



F119894(119905) = minus

[1 (119890119905minus 1)](119894+1)

[1 (119890119905minus 1)](119894)

= minus

(suminfin

119896=1119890minus119896119905)

(119894+1)

(suminfin

119896=1119890minus119896119905)(119894)

= minus

suminfin

119896=1(minus1)119894+1119896119894+1119890minus119896119905

suminfin

119896=1(minus1)119894119896119894119890minus119896119905

=

suminfin

119896=1119896119894+1119890minus119896119905

suminfin

119896=1119896119894119890minus119896119905

(19)

Let 119886119896= 119896119894+1 119887119896= 119896119894 and 119906

119896(119905) = 119890

minus119896119905 Then 119886119896119887119896= 119896 is an




lim119905rarrinfin

F119894(119905) = lim119905rarrinfin

suminfin

119896=1119896119894+1119890minus119896119905

suminfin

119896=1119896119894119890minus119896119905


1 + suminfin

119896=2119896119894+1119890(1minus119896)119905

1 + suminfin

119896=2119896119894119890(1minus119896)119905

= 1

(20)


119894(119905) =


119894(119905) has the same





G0(119905) =

1

119890119905minus 1

= 1198650(119905) (21)


4 Some Applications


tion

Li119904(119911) =

infin

sum

119896=1

119911119896

119896119904

(22)



infin


that

119865119894(119905) = Li

minus119894(119890minus119905) 119866

119894+1(119905) = Li

minus(119894+1)(119890minus119905) (23)


119904(119911) as follows






[Liminus119894(119890minus119905)]

(119896)

= (minus1)119896Liminus(119894+119896)

(119890minus119905) (24)


F119894(119905) = Li


minus119894(119890minus119905)

G119894(119905) =


F119894+1

(119905) 119894 ge 1

(25)





minus(119894+1)(119890minus119905) (26)



F119894(119905) =

Liminus(119894+1)

(119890minus119905)

Liminus119894(119890minus119905)

G119894(119905) =

F119894(119905) 119894 ge 1

Liminus119894(119890minus119905)

1 + Li0(119890minus119905)

119894 = 0

(27)


lim119905rarrinfin

F119894(119905) = 1 lim

119905rarrinfin

G0(119905) = 0 lim

119905rarrinfin

G119894+1

(119905) = 1

(28)









P119894(119905) = Li

minus(119894+1)(1

119905


119905

)

Q119894(119905) =

Liminus119894(1

119905

) minus Li0(1

119905

) + 1 119894 = 0

P119894+1

(119905) 119894 ge 1

(29)









minus119894(119890minus119905) and 119892(119905)






5 A Conjecture


G0(119905) = F

0(119905) minus 1 =

1

119890119905minus 1

= 1198650(119905)

G1(119905) = F

1(119905) = 1 +

2

119890119905minus 1

= 1 + 21198650(119905)

(30)




(0infin)



Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














sum

119896=0

119886119896119906(ℓ)

119896(119905)

infin

sum

119896=0

119887119896119906(ℓ)

119896(119905) ℓ = 0 1 (13)





119891 (119905) =

suminfin

119896=0119886119896119906119896(119905)

suminfin

119896=0119887119896119906119896(119905)

(14)


119896(119905)119906119896(119905) forms a de-






1198650(119905) = 119866

0(119905) minus 1 =

infin

sum

ℓ=1

119890minusℓ119905

119865119894(119905) = 119866

119894(119905) =

infin

sum

ℓ=1

ℓ119894119890minusℓ119905

(15)


119894(119905) for 119894 isin 0 cup N are


0(119905) and119866

0(119905) and


[ln1198650(119905)]1015840

=dd119905(ln 1

119890119905minus 1

) = minus119890119905

119890119905minus 1

= minus [1198650(119905) + 1]

[ln1198660(119905)]1015840

=dd119905(ln 1


0(119905)

(16)




0(119905) on




119865119894+1

(119905) minus 119865119894(119905)

= (minus1)119894+1

[(1

119890119905minus 1

)

(119894+1)

+ (1

119890119905minus 1

)

(119894)

]

= (minus1)119894+1 [

[

(

infin

sum

ℓ=1

119890minusℓ119905)

(119894+1)

+ (

infin

sum

ℓ=1

119890minusℓ119905)

(119894)

]

]

= (minus1)119894+1

[

infin

sum

ℓ=1

(minus1)119894+1ℓ119894+1119890minusℓ119905

+

infin

sum

ℓ=1

(minus1)119894ℓ119894119890minusℓ119905]

=

infin

sum

ℓ=2

(ℓ minus 1) ℓ119894119890minusℓ119905

gt 0

(minus1)119899[119865119894+1

(119905) minus 119865119894(119905)](119899)

=

infin

sum

ℓ=2

(ℓ minus 1) ℓ119899+119894119890minusℓ119905

gt 0

(17)


119894+1(119905) minus 119865

119894(119905)


the functions

G0(119905) = 119866

1(119905) minus 119866

0(119905) + 1 = 119865

1(119905) minus 119865

0(119905)

G119894+1

(119905) = 119866119894+2

(119905) minus G119894+1

(119905) = 119865119894+2

(119905) minus 119865119894+1

(119905)

(18)


119894(119905)



F119894(119905) = minus

[1 (119890119905minus 1)](119894+1)

[1 (119890119905minus 1)](119894)

= minus

(suminfin

119896=1119890minus119896119905)

(119894+1)

(suminfin

119896=1119890minus119896119905)(119894)

= minus

suminfin

119896=1(minus1)119894+1119896119894+1119890minus119896119905

suminfin

119896=1(minus1)119894119896119894119890minus119896119905

=

suminfin

119896=1119896119894+1119890minus119896119905

suminfin

119896=1119896119894119890minus119896119905

(19)

Let 119886119896= 119896119894+1 119887119896= 119896119894 and 119906

119896(119905) = 119890

minus119896119905 Then 119886119896119887119896= 119896 is an




lim119905rarrinfin

F119894(119905) = lim119905rarrinfin

suminfin

119896=1119896119894+1119890minus119896119905

suminfin

119896=1119896119894119890minus119896119905


1 + suminfin

119896=2119896119894+1119890(1minus119896)119905

1 + suminfin

119896=2119896119894119890(1minus119896)119905

= 1

(20)


119894(119905) =


119894(119905) has the same





G0(119905) =

1

119890119905minus 1

= 1198650(119905) (21)


4 Some Applications


tion

Li119904(119911) =

infin

sum

119896=1

119911119896

119896119904

(22)



infin


that

119865119894(119905) = Li

minus119894(119890minus119905) 119866

119894+1(119905) = Li

minus(119894+1)(119890minus119905) (23)


119904(119911) as follows






[Liminus119894(119890minus119905)]

(119896)

= (minus1)119896Liminus(119894+119896)

(119890minus119905) (24)


F119894(119905) = Li


minus119894(119890minus119905)

G119894(119905) =


F119894+1

(119905) 119894 ge 1

(25)





minus(119894+1)(119890minus119905) (26)



F119894(119905) =

Liminus(119894+1)

(119890minus119905)

Liminus119894(119890minus119905)

G119894(119905) =

F119894(119905) 119894 ge 1

Liminus119894(119890minus119905)

1 + Li0(119890minus119905)

119894 = 0

(27)


lim119905rarrinfin

F119894(119905) = 1 lim

119905rarrinfin

G0(119905) = 0 lim

119905rarrinfin

G119894+1

(119905) = 1

(28)









P119894(119905) = Li

minus(119894+1)(1

119905


119905

)

Q119894(119905) =

Liminus119894(1

119905

) minus Li0(1

119905

) + 1 119894 = 0

P119894+1

(119905) 119894 ge 1

(29)









minus119894(119890minus119905) and 119892(119905)






5 A Conjecture


G0(119905) = F

0(119905) minus 1 =

1

119890119905minus 1

= 1198650(119905)

G1(119905) = F

1(119905) = 1 +

2

119890119905minus 1

= 1 + 21198650(119905)

(30)




(0infin)



Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












G0(119905) =

1

119890119905minus 1

= 1198650(119905) (21)


4 Some Applications


tion

Li119904(119911) =

infin

sum

119896=1

119911119896

119896119904

(22)



infin


that

119865119894(119905) = Li

minus119894(119890minus119905) 119866

119894+1(119905) = Li

minus(119894+1)(119890minus119905) (23)


119904(119911) as follows






[Liminus119894(119890minus119905)]

(119896)

= (minus1)119896Liminus(119894+119896)

(119890minus119905) (24)


F119894(119905) = Li


minus119894(119890minus119905)

G119894(119905) =


F119894+1

(119905) 119894 ge 1

(25)





minus(119894+1)(119890minus119905) (26)



F119894(119905) =

Liminus(119894+1)

(119890minus119905)

Liminus119894(119890minus119905)

G119894(119905) =

F119894(119905) 119894 ge 1

Liminus119894(119890minus119905)

1 + Li0(119890minus119905)

119894 = 0

(27)


lim119905rarrinfin

F119894(119905) = 1 lim

119905rarrinfin

G0(119905) = 0 lim

119905rarrinfin

G119894+1

(119905) = 1

(28)









P119894(119905) = Li

minus(119894+1)(1

119905


119905

)

Q119894(119905) =

Liminus119894(1

119905

) minus Li0(1

119905

) + 1 119894 = 0

P119894+1

(119905) 119894 ge 1

(29)









minus119894(119890minus119905) and 119892(119905)






5 A Conjecture


G0(119905) = F

0(119905) minus 1 =

1

119890119905minus 1

= 1198650(119905)

G1(119905) = F

1(119905) = 1 +

2

119890119905minus 1

= 1 + 21198650(119905)

(30)




(0infin)



Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 A Conjecture


G0(119905) = F

0(119905) minus 1 =

1

119890119905minus 1

= 1198650(119905)

G1(119905) = F

1(119905) = 1 +

2

119890119905minus 1

= 1 + 21198650(119905)

(30)




(0infin)



Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article complete monotonicity of functions

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