A certain function class related to the class of logarithmically completely monotonic functions

Mathematical and Computer Modelling 49 (2009) 2073–2079

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A certain function class related to the class of logarithmically completely monotonic functions Senlin Guo a , H.M. Srivastava b,∗ a

Department of Mathematics, Zhongyuan University of Technology, Zhengzhou, Henan 450007, People’s Republic of China

b

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

article

info

Article history: Received 12 September 2008 Received in revised form 31 December 2008 Accepted 5 January 2009

a b s t r a c t In this article, we introduce and investigate the notion of strongly logarithmically completely monotonic functions. Among other results, we present some properties and relationships involving this function class and several closely-related function classes. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Completely monotonic functions Logarithmically completely monotonic functions Strongly completely monotonic functions Strongly logarithmically completely monotonic functions Multinomial coefficients

1. Introduction Throughout the present investigation, we denote by N the set of all positive integers and set

N0 := N ∪ {0} and R+ := (0, ∞). We suppose also that I + is an open interval contained in R+ and I 0 is the interior of the interval I ⊂ R. We first recall each of the following definitions. Definition A (See, for example, [1,2]). A function f is said to be completely monotonic on an interval I if f ∈ C (I ) has derivatives of all orders on I 0 and, for all n ∈ N0 ,

(−1)n f (n) (x) = 0 (x ∈ I 0 ; n ∈ N0 ). The class of all completely monotonic functions on I is denoted by CM (I ). Definition B (See [3,4]). A function f is said to be logarithmically completely monotonic on an interval I if f >0





f ∈ C (I )

has derivatives of all orders on I 0 and, for all n ∈ N,

(−1)n [ln f (x)](n) = 0 (x ∈ I 0 ; n ∈ N). ∗

Corresponding author. E-mail addresses: [email protected] (S. Guo), [email protected] (H.M. Srivastava).

0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.01.002

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The set of all logarithmically completely monotonic functions on I is denoted here by LCM (I ) (see also each of the recent investigations on this subject by Guo et al. in [5 to 7]). Horn [8] proved, in terms of logarithmically completely monotonic functions, that f ∈ LCM (R+ ) ⇐⇒ f 6≡ 0 and f ∈ CM (R+ )

p n

(n ∈ N).

A function f such that f ∈ CM (R+ )

p n

(n ∈ N)

is called infinitely divisible completely monotonic (cf. [8]). From Horn’s investigation in [8], we have LCM (R+ ) ⊂ CM (R+ ), which was rederived in [4] by following a considerably different approach from that of Horn [8]. Definition C (See [9]). A function f is said to be strongly completely monotonic on I + if, for all n ∈ N0 , the following functions:

(−1)n xn+1 f (n) (x) (n ∈ N0 ) are nonnegative and decreasing on I + . The class of all such functions is denoted by SCM (I + ). Clearly, we have SCM (I + ) ⊂ CM (I + ). Let α = 1. Then fα (x) := x−α ∈ LCM (I + ) ∩ SCM (I + ). Therefore, we have LCM (I + ) ∩ SCM (I + ) 6= ∅. Motivated by the above concepts and definitions, we introduce the following notion. Definition 1. A function f is said to be strongly logarithmically completely monotonic on I + if f > 0 (f ∈ I + ) and, for all n ∈ N, the following functions:

(−1)n xn+1 [ln f (x)](n)

(n ∈ N)

are nonnegative and decreasing on I + . Such a function class is denoted by SLCM (I + ). For example, we have e1/x ∈ SLCM (R+ ). But e−x ∈ LCM (R+ ) \ SLCM (R+ ). From the above definitions, we also find that SLCM (I + ) ⊂ LCM (I + ). In order to simplify the statements of our main results, we shall use the following terminology. Definition 2. A function f is said to be almost strongly completely monotonic on I + if, for all n ∈ N, the following functions:

(−1)n xn+1 f (n) (x) (n ∈ N) are nonnegative and decreasing on I + . The above-defined class of almost strongly completely monotonic functions on I + is denoted henceforth by ASCM (I + ). The object of the present investigation is to prove several inclusion properties and other relationships associated with the various function classes which are introduced above.

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2. The main results Our main results are now stated as Theorems 1 to 3 below. Theorem 1. A strongly logarithmically completely monotonic function on I + is almost strongly completely monotonic on I + , that is, SLCM (I + ) ⊂ ASCM (I + ). Remark 1. It is easy to see that SLCM (I + ) ∪ SCM (I + ) ⊂ CM (I + )

(1)

and that SCM (I + ) ⊂ ASCM (I + ).

(2)

From Theorem 1 and equation (2), we find that SLCM (I + ) ∪ SCM (I + ) ⊂ ASCM (I + ).

(3)

Moreover, from (1) and (3), we see that both CM (I ) and ASCM (I ) contain the following function class: +

+

SLCM (I + ) ∪ SCM (I + ). The following examples show that CM (I + ) and ASCM (I + ) do not contain each other: 1

− 1 ∈ ASCM (R+ ) \ CM (R+ )

x2 and 1

√ ∈ CM (R+ ) \ ASCM (R+ ). x

Theorem 2 below reveals an important relationship between SLCM (R+ ) and SCM (R+ ). Theorem 2. The following relationship holds true: SLCM (R+ ) ∩ SCM (R+ ) = ∅, that is, a strongly logarithmically completely monotonic function on R+ cannot be strongly completely monotonic on R+ ; or, equivalently, a strongly completely monotonic function on R+ cannot be strongly logarithmically completely monotonic on R+ . Theorem 3. (1) Suppose that f ∈ C (I ),

f > 0 and

f 0 ∈ CM (I 0 ).

Then 1

∈ LCM (I ). f (2) Suppose that f ∈ C (I + ),

f > 0 and

f 0 ∈ SCM (I + ).

If xf 0 (x) = f (x)

(x ∈ I + ),

then 1

∈ SLCM (I + ). f (3) Suppose that f ∈ C (I + ),

f > 0 and

Then 1 f

∈ SLCM (I + ).

Remark 2. The following condition: xf 0 (x) = f (x)

(x ∈ I + )

− f ∈ ASCM (I + ).

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in Theorem 3(2) cannot be dropped. For example, if we let f (x) := ln x

and I + := (e, ∞),

then 1

f 0 (x) =

x

∈ SCM (I + )

and the following condition: xf 0 (x) = f (x)

(x ∈ I + )

is not satisfied. It is easy to verify that the function:

 

(−1) x

1 2

0

1

ln

=

ln x

x ln x

is strictly increasing on I + . Therefore, we have 1 f (x)

=

1 ln x

6∈ SLCM (I + ).

3. Proofs of Theorems 1 to 3 We need the following lemma to prove the main results stated already in Section 2. Lemma 1 (See [10, p. 21]). Suppose that each of the following functions: y = y(x)

(x ∈ I1 ) and x = ϕ(t ) (t ∈ I )

is differentiable n times. Also let the range:

R(ϕ) ⊂ I1 . Then, for t ∈ I , dn y dt n



X

=

(i1 ,...,in )∈Λn



n i1 , . . . , in

dxm

dm y x=ϕ(t )

( i ) n Y ϕ (j) (t ) j · j! j =1

(m = i1 + · · · + in ; t ∈ I ), where





n i1 , . . . , in

=

n! i1 ! · · · in !

denotes the multinomial coefficient and

Λn := (i1 , . . . , in ) i1 , . . . , in ∈ N0 and (

n X

) kik = n

.

(4)

k=1

Proof of Theorem 1. Suppose that f ∈ SLCM (I + ). Also let g (x) = ln f (x)

(x ∈ I + ).

Then, by definition, the following functions:

(−1)` x`+1 g (`) (x) (` ∈ N)

(5)

are nonnegative and decreasing on I n

d f dxn

=

X (i1 ,...,in )∈Λn



+

and for all ` ∈ N. Thus, by Lemma 1, for all n ∈ N, we have



n i1 , . . . , in

e

g (x)

( i ) n Y g (j) (x) j , · j! j =1

where Λn is defined by (4). For any (i1 , . . . , in ) ∈ Λn , it is easily observed that

(−1)n xn+1 =

(−1)i1 +2i2 +···+nin x2i1 +3i2 +···+(n+1)in xi1 +···+in −1

,

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which readily yields n n+1 (n)

(−1) x

f

X

(x) =

(i1 ,...,in )∈Λn



n



i1 , . . . , in

eg (x) xi1 +···+in −1

( i ) n Y (−1)j xj+1 g (j) (x) j · . j! j=1

(6)

Since (i1 , . . . , in ) ∈ Λn , we have i1 + · · · + in = 1. Now, by setting ` = 1 in the condition (5), we see that g 0 (x) 5 0 (x ∈ I + ), that is, g (x) is decreasing on I + . Therefore, for (i1 , . . . , in ) ∈ Λn , the following function: eg (x) x1−(i1 +···+in ) is nonnegative and decreasing on I + . Also, by the condition (5), each factor in (6) following the factor: eg (x) x1−(i1 +···+in ) is nonnegative and decreasing on I + . We, therefore, conclude that, for all n ∈ N, the following functions:

(−1)n xn+1 f (n) (x) (n ∈ N) are nonnegative and decreasing on I + . By definition, f ∈ ASCM (I + ). The proof of Theorem 1 is thus completed.



Proof of Theorem 2. It is sufficient to show that, if f ∈ SLCM (R ), then +

f 6∈ SCM (R+ ). For this purpose, we suppose that f ∈ SLCM (R+ ). We also let g (x) = ln f (x)

(x ∈ R+ ).

Then, by definition, the function −x2 g 0 (x) is decreasing on R+ . Therefore, we have lim {−x2 g 0 (x)} 5 −g 0 (1).

(7)

x→∞

We now claim that f (x) 6∈ SCM (R+ ). If f (x) were in SCM (R+ ), then the function xf (x) would be decreasing on R+ , that is,

[xf (x)]0 5 0 (x ∈ R+ ).

(8)

Since

[xf (x)]0 = f (x) + xf 0 (x) = eg (x) + xeg (x) g 0 (x)
= eg (x) 1 + xg 0 (x) , we find from (8) that 1 + xg 0 (x) 5 0 (x ∈ R+ ),

(9)

which obviously yields the following inequality:

−x2 g 0 (x) = x (x ∈ R+ ). Therefore, we have lim {−x2 g 0 (x)} = ∞,

x→∞

which, in view of (7), leads us to a contradiction. Our claim has thus been proved.



Proof of Theorem 3(1). Without any loss of generality, we may assume that I is an open interval. We also let h(x) := ln



1



f (x)

= − ln f (x) (x ∈ I ).

It is easy to verify that

(ln x)(k) =

(−1)k−1 (k − 1)! xk

(k ∈ N).

(10)

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Since f 0 ∈ CM (I ), we easily obtain

(−1)` [f 0 (x)](`) = (−1)` f (`+1) (x) = 0 (x ∈ I ; ` ∈ N0 ).

(11)

By Lemma 1 and (10), we deduce for n ∈ N that

( i ) n (m − 1)! Y f (j) (x) j (−1) h (x) = (−1) (−1) · · i1 , . . . , in [f (x)]m j=1 j! (i1 ,...,in )∈Λn (    i ) n X (m − 1)! Y n (−1)j−1 f (j) (x) j 2m = (−1) · i1 , . . . , in [f (x)]m j=1 j! (i ,...,i )∈Λ n (n)

X

n +1

n

1

m−1





n

n

(m = i1 + i2 + · · · + in = 1),

(12)

where Λn is defined, as elsewhere, by (4). By combining (11) and (12), we get

(−1)n h(n) (x) = 0 (x ∈ I , n ∈ N). Hence 1 f

∈ LCM (I ),

which evidently completes the proof of Theorem 3(1).



Proof of Theorem 3(2). Let



h(x) := ln



1 f ( x)

= − ln f (x) (x ∈ I + ).

Then, since f 0 ∈ SCM (I + ), the following functions:

(−1)` x`+1 f (`+1) (x) (` ∈ N0 )

(13)

are nonnegative and decreasing on I

for all ` ∈ N0 . By Lemma 1 and equation (10), we find for n ∈ N that

( i ) n (m − 1)! Y f (j) (x) j h (x) = (−1) x (−1) · · i1 , . . . , in [f (x)]m j=1 j! (i1 ,...,in )∈Λn (    i ) n X (−1)j−1 xj f (j) (x) j n x(m − 1)! Y 2m = (−1) · i1 , . . . , in [f (x)]m j! j=1 (i ,...,i )∈Λ

n n+1 (n)

(−1) x

+

n +1 n +1

n

1

X

m−1



n



n

(m = i1 + · · · + in = 1),

(14)

where Λn is defined by (4). Since xf 0 (x) = f (x)

(x ∈ I + ),

the following function: x

(15)

f (x) is positive and decreasing on I + . Thus, by setting ` = 0 in (13), we find that f 0 (x) = 0 (x ∈ I + ),

(16)

that is, that f (x) is increasing on I . Making use of (15) and (16), as well as the fact that +

x

[f (x)]

m

=

1

[f (x)]

x m−1

f ( x)

(m ∈ N),

we see that the following functions: x

[f (x)]m

(m ∈ N)

(17)

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are nonnegative and decreasing on I + for all m ∈ N. In light of the observations in (13) and (17), we conclude from (14) that the following functions:

(−1)n xn+1 h(n) (x) (n ∈ N) are nonnegative and decreasing on I + for n ∈ N. Hence, we have 1 f

∈ SLCM (I + ),

which completes the proof of Theorem 3(2).



Proof of Theorem 3(3). We first let h(x) := ln



1 f (x)



= − ln f (x) (x ∈ I + ).

Then, since −f ∈ ASCM (I + ), the following functions:

(−1)`+1 x`+1 f (`) (x) (` ∈ N)

(18)

are nonnegative and decreasing on I

for ` ∈ N. By Lemma 1 and equation (10), we find for n ∈ N that

( i ) n (m − 1)! Y f (j) (x) j h (x) = (−1) x (−1) · · i1 , . . . , in [f (x)]m j=1 j! (i1 ,...,in )∈Λn ( )    i n X (−1)j+1 xj+1 f (j) (x) j n (m − 1)! Y · = m − 1 m [f (x)] j! i1 , . . . , in x j =1 (i ,...,i )∈Λ

n n+1 (n)

(−1) x

+

X

n +1 n +1

1

n

m−1



n



n

(m = i1 + · · · + in = 1),

(19)

where Λn is defined by (4). By setting ` = 1 in (18), we get f 0 (x) = 0 (x ∈ I + ),

(20)

that is, the function f (x) is increasing on I . Since m = 1, in view of (18) and (20), we conclude from (19) that the following functions: +

(−1)n xn+1 h(n) (x) (n ∈ N) are nonnegative and decreasing on I + for n ∈ N. Hence 1 f

∈ SLCM (I + ),

which evidently completes the proof of Theorem 3(3).



We conclude this investigation by remarking that several corollaries and consequences of each of Theorems 1 to 3 can be deduced fairly easily. The details involved are being left as an exercise for the interested reader. Acknowledgements The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] D.S. Mitrinović, J.E. Pečarić, (With an Appendix by M. R. Tasković), Monotone Functions and Their Inequalities, Series on Mathematical Problems and Expositions, vol. 17, Naučna Knjiga, Belgrade, 1990 (Serbo-Croatian). [2] D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, New Jersey, 1966. [3] R.D. Atanassov, U.V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988) 21–23. [4] F. Qi, C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004) 603–607. [5] S. Guo, F. Qi, H.M. Srivastava, Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic, Integral Transforms Spec. Funct. 18 (2007) 819–826. [6] S. Guo, F. Qi, H.M. Srivastava, Supplements to a class of logarithmically completely monotonic functions associated with the gamma function, Appl. Math. Comput. 197 (2008) 768–774. [7] S. Guo, H.M. Srivastava, A class of logarithmically completely monotonic functions, Appl. Math. Lett. 21 (2008) 1134–1141. [8] R.A. Horn, On infinitely divisible matrices, kernels, and functions, Z. Wahrscheinlichkeitstheor. Verwandte. Geb. 8 (1967) 219–230. [9] S.Y. Trimble, J. Wells, F.T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal. 20 (1989) 1255–1259. [10] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, sixth ed., Academic Press, New York, 2000.