Contents lists available at ScienceDirect ... · JournalofComputationalandAppliedMathematics233(2010)2149 2160 Contents lists available at ScienceDirect JournalofComputationalandApplied
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Journal of Computational and Applied Mathematics 233 (2010) 2149–2160
Contents lists available at ScienceDirect
Journal of Computational and AppliedMathematics
journal homepage: www.elsevier.com/locate/cam
Complete monotonicity of some functions involvingpolygamma functionsFeng Qi a,∗, Senlin Guo b, Bai-Ni Guo aa Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, Chinab Department of Mathematics, Zhongyuan University of Technology, Zhengzhou City, Henan Province, 450007, China
a r t i c l e i n f o
Article history:Received 5 January 2009Received in revised form 25 September2009
In the present paper, we establish necessary and sufficient conditions for the functionsxα |ψ (i)(x + β)| and α|ψ (i)(x + β)| − x|ψ (i+1)(x + β)| respectively to be monotonic andcompletely monotonic on (0,∞), where i ∈ N, α > 0 and β ≥ 0 are scalars, and ψ (i)(x)are polygamma functions.
Recall [1, Chapter XIII] and [2, Chapter IV] that a function f (x) is said to be completely monotonic on an interval I ⊆ R iff (x) has derivatives of all orders on I and
0 ≤ (−1)kf (k)(x) <∞ (1.1)
holds for all k ≥ 0 on I . This definition was introduced in 1921 in [3], who called such functions ‘‘total monoton’’.The celebrated Bernstein–Widder Theorem [2, p. 161] states that a function f (x) is completely monotonic on (0,∞) if
2150 F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160
whereµ is a nonnegative measure on [0,∞) such that the integral (1.2) converges for all x > 0. This means that a functionf (x) is completely monotonic on (0,∞) if and only if it is a Laplace transform of the measure µ.The most important properties of completely monotonic functions can be found in [1, Chapter XIII], [2, Chapter IV], [4,5]
and the related references therein.The completely monotonic functions have applications in different branches of mathematical sciences. For example,
they play some role in combinatorics [6], numerical and asymptotic analysis [7,8], physics [9,10], potential theory [11], andprobability theory [12,10,13].
1.2
It is well known [14–16] that the classical Euler gamma function may be defined for x > 0 by
0(x) =∫∞
0tx−1e−tdt. (1.3)
The logarithmic derivative of 0(x), denoted by ψ(x) = 0′(x)0(x) , is called psi function, and ψ
(k)(x) for k ∈ N are calledpolygamma functions.It should be common knowledge [14–16] that the special functions 0(x), ψ(x) and ψ (i)(x) for i ∈ N are important and
basic and that they have much extensive applications in mathematical sciences.
1.3
In [17, Lemma 1], it was shown that the functions xc∣∣ψ (k)(x)
∣∣ for k ∈ N and c ∈ R are strictly decreasing (or strictlyincreasing, respectively) on (0,∞) if and only if c ≤ k (or c ≥ k+ 1, respectively).In [18, Theorem 4.14], it was obtained that the function xc
∣∣ψ (k)(x)∣∣ for k ∈ N and c ∈ R is strictly convex on (0,∞) if
and only if either c ≤ k, or c = k+ 1, or c ≥ k+ 2. In [18, Remark 4.15], it was pointed out that there does not exist a realnumber c such that the function xc
∣∣ψ (k)(x)∣∣ for k ∈ N is concave on (0,∞).
In [18, Lemma 2.2] and [19, Lemma 5], the functions xα∣∣ψ (i)(x + 1)
∣∣ for i ∈ N are proved to be strictly increasing (orstrictly decreasing, respectively) on (0,∞) if and only if α ≥ i (or α ≤ 0, respectively).In [20, Lemma 2.1], the function xψ ′(x+ a) is proved to be strictly increasing on [0,∞) for a ≥ 1.Motivated by the above results, the first and third authors considered in [21] themonotonicity of amore general function
xα∣∣ψ (i)(x+ β)
∣∣ and the complete monotonicity of several related functions as follows: For i ∈ N, α > 0 and β ≥ 0,
(1) the function xα∣∣ψ (i)(x+β)
∣∣ is strictly increasing on (0,∞) if (α, β) ∈ {α ≥ i, 12 ≤ β < 1}∪{α ≥ i, β ≥ α−i+12
}∪{α ≥
i+ 1, β ≤ α−i+12
}and only if α ≥ i;
(2) the function αx
∣∣ψ (i)(x)∣∣− ∣∣ψ (i+1)(x)
∣∣ is completely monotonic on (0,∞) if and only if α ≥ i+ 1;(3) the function
∣∣ψ (i+1)(x)∣∣− α
x
∣∣ψ (i)(x)∣∣ is completely monotonic on (0,∞) if and only if 0 < α ≤ i;
(4) the function αx
∣∣ψ (i)(x+ 1)∣∣−∣∣ψ (i+1)(x+ 1)
∣∣ is completely monotonic on (0,∞) if and only if α ≥ i;(5) the function αx
∣∣ψ (i)(x+β)∣∣−∣∣ψ (i+1)(x+β)
∣∣ on (0,∞) is completelymonotonic if (α, β) ∈ {α ≥ i+1, β ≤ α−i+12
}∪{i ≤
α ≤ i+ 1, β ≥ α−i+12
}∪{i ≤ α ≤ (i+1)(i+4β−2)
i+2β , 12 ≤ β < 1}and only if α ≥ i;
(6) the function α∣∣ψ (i)(x + β)
∣∣−x∣∣ψ (i+1)(x + β)∣∣ is completely monotonic on (0,∞) if (α, β) ∈ {i ≤ α ≤ i + 1, β ≥
α−i+12
}∪{α ≥ i+ 1, β ≤ α−i+1
2
}and only if α ≥ i.
1.4
The first aimof this paper is to present necessary and sufficient conditions for the function xα∣∣ψ (i)(x+β)
∣∣ to bemonotonicon (0,∞), which can be summarized as the following Theorem 1.
Theorem 1. Let i ∈ N, α ∈ R and β ≥ 0.
(1) The function xα∣∣ψ (i)(x)
∣∣ is strictly increasing (or strictly decreasing, respectively) on (0,∞) if and only if α ≥ i+1 (or α ≤ i,respectively).
(2) For β ≥ 12 , the function x
α∣∣ψ (i)(x+ β)
∣∣ is strictly increasing on [0,∞) if and only if α ≥ i.(3) Let δ : (0,∞)→
(0, 12
)be defined by
δ(t) =et(t − 1)+ 1(et − 1)2
(1.4)
F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160 2151
and δ−1 :(0, 12
)→ (0,∞) stand for the inverse function of δ. If 0 < β < 1
2 and
α ≥ i+ 1−[eδ−1(β)
eδ−1(β) − 1+ β − 1
]δ−1(β), (1.5)
then the function xα∣∣ψ (i)(x+ β)
∣∣ is strictly increasing on (0,∞).As a by-product of the proof of Theorem1, lower and upper bounds for infinite serieswhose coefficients involve Bernoulli
numbers may be derived as follows.
Corollary 1. Let 0 < β < 12 and δ
−1 be the inverse function of δ defined by (1.4). Then the following inequalities hold:
12>
∞∑k=1
B2kt2k−1
(2k− 1)!> 0, (1.6)
t2>
∞∑k=0
B2k+2t2k+2
(2k+ 2)!> max
{0,t2− 1
}, (1.7)
∞∑k=0
B2k+2t2k+2
(2k+ 2)!>
(12− β
)t +
[eδ−1(β)
eδ−1(β) − 1− β + 1
]δ−1(β)− 1, (1.8)
where t ∈ (0,∞) and Bn for n ≥ 0 represent Bernoulli numbers which may be defined [14–16] by
xex − 1
=
∞∑n=0
Bnn!xn = 1−
x2+
∞∑j=1
B2jx2j
(2j)!, |x| < 2π. (1.9)
1.5
The second aim of this paper is to establish necessary and sufficient conditions for the function α∣∣ψ (i)(x + β)
∣∣−x∣∣ψ (i+1)(x+ β)
∣∣ to be completely monotonic on (0,∞), which may be stated as the following Theorem 2.Theorem 2. Let i ∈ N, α ∈ R and β ≥ 0.
(1) The function
α∣∣ψ (i)(x)
∣∣−x∣∣ψ (i+1)(x)∣∣ (1.10)
is completely monotonic on (0,∞) if and only if α ≥ i+ 1.(2) The negative of the function (1.10) is completely monotonic on (0,∞) if and only if α ≤ i.(3) If β ≥ 1
2 , then the function
α∣∣ψ (i)(x+ β)
∣∣−x∣∣ψ (i+1)(x+ β)∣∣ (1.11)
is completely monotonic on (0,∞) if and only if α ≥ i.(4) If 0 < β < 1
2 and the inequality (1.5) is valid, then the function (1.11) is completely monotonic on (0,∞).
As immediate consequences of Theorem 2, the following corollary is obtained.
Corollary 2. Let i ∈ N, α ∈ R and β ≥ 0.
(1) The functionα
x
∣∣ψ (i)(x)∣∣−∣∣ψ (i+1)(x)
∣∣ (1.12)
is completely monotonic on (0,∞) if and only if α ≥ i+ 1.(2) The negative of the function (1.12) is completely monotonic on (0,∞) if and only if α ≤ i.(3) If β ≥ 1
2 , then the function
α
x
∣∣ψ (i)(x+ β)∣∣−∣∣ψ (i+1)(x+ β)
∣∣ (1.13)
is completely monotonic on (0,∞) if and only if α ≥ i.(4) If 0 < β < 1
2 and the inequality (1.5) holds true, then the function (1.13) is completely monotonic on (0,∞).
2152 F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160
2. Remarks and applications
Before proving the above theorems and corollaries, we give several remarks about Theorem 1, Theorem 2 and theirapplications.
Remark 1. Since limβ→0+ [βδ−1(β)] = 0 and that the inverse function δ−1 is decreasing from(0, 12
)onto (0,∞), we claim
that
0 <[eδ−1(β)
eδ−1(β) − 1+ β − 1
]δ−1(β) < 1, β ∈ (0, 1). (2.1)
Indeed, if replacing δ−1(β) by s, the middle term in (2.1) becomes s2es
(es−1)2which is decreasing from (0,∞) onto (0, 1). This
implies that the condition (1.5) is not only sufficient but also necessary in Theorems 1 and 2.
Remark 2. As mentioned in Section 1.3, some conclusions in Theorem 1 have been applied in nearby fields.
(1) The first conclusion in Theorem1wasutilized in [17, Theorem2] to obtain a functional inequality concerningpolygammafunctions: For k ≥ 1 and n ≥ 2, the inequality∣∣ψ (k)(M [r]n (xν; pν))∣∣ ≤ M [s]n (∣∣ψ (k)(xν)
∣∣; pν) (2.2)
holds if and only if either r ≥ 0 and s ≥ − rk+1 or r < 0 and s ≥ −
rk , where xν > 0 and pν > 0 with
∑nν=1 pν = 1, and
M [t]n (xν; pν) =
( n∑ν=1
pνxtν
)1/t, t 6= 0
n∏ν=1
xpνν , t = 0(2.3)
stands for the discrete weighted power means.(2) Some applications of the first two conclusions in Theorem 1 were carried out in [18] as follows.(a) The first conclusion in Theorem 1 was applied in [18, Theorem 4.9] to obtain that the inequalities(
1+α
x+ s
)n<ψ (n)(x+ s)ψ (n)(x+ 1)
<
(1+
β
x+ s
)n(2.4)
hold for n ∈ N and s ∈ (0, 1)with the best possible constants
α = 1− s and β = s[ψ (n)(s)ψ (n)(1)
]1/n− s. (2.5)
(b) The special cases for β = 1 of the third conclusion in Theorem 1 was employed in [18, Theorem 4.8] to derive thatthe inequalities
(n− 1)! exp[α
x− nψ(x)
]<∣∣ψ (n)(x)
∣∣ < (n− 1)! exp[β
x− nψ(x)
](2.6)
hold for n ∈ N and x > 0 if and only if α ≤ −n and β ≥ 0.(c) Moreover, the convexity of xc
∣∣ψ (k)(x)∣∣was used in [18, Theorem 4.16] to establish that the double inequalities
α
(1x−1y
)< xn
∣∣ψ (n)(x)∣∣− yn∣∣ψ (n)(y)
∣∣ < β
(1x−1y
)(2.7)
hold for n ∈ N and y > x > 0 with the best possible constants α = n!2 and β = n!.
(3) The special cases for β = 1 of the third conclusion in Theorem 1, the monotonic properties of the functions
on [0,∞), were used in [19,22] to establish the monotonic, logarithmically convex, completely monotonic propertiesof the functions
[0(1+ x)]y
0(1+ xy)and
0(1+ y)[0(1+ x)]y
0(1+ xy)(2.8)
or their first and second logarithmic derivatives.(4) The special case for β ≥ 1 and i = 1 of the third conclusion in Theorem 1 was employed in [20, Theorem 3.1] to revealthe subadditive property of the function ψ(a+ ex) on (−∞,∞).
Remark 3. Let us recall the following four definitions.
F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160 2153
Definition 1 ([23, Definition 1.1]). Let f be a positive functionwhich has derivatives of all orders on an interval I . If [ln f (x)](k)for some nonnegative integer k is completely monotonic on I , but [ln f (x)](k−1) is not completely monotonic on I , then f issaid to be a logarithmically completely monotonic function of kth order on I .
Definition 2 ([23, Definition 1.2]). Let f be a positive functionwhich has derivatives of all orders on an interval I . If [ln f (x)](k)for some nonnegative integer k is absolutely monotonic on I , but [ln f (x)](k−1) is not absolutely monotonic on I , then f issaid to be a logarithmically absolutely monotonic function of kth order on I .
Definition 3 ([23, Definition 1.3]). A positive function f which has derivatives of all orders on an interval I is said to belogarithmically absolutely convex on I if [ln f (x)](2k) ≥ 0 on I for k ∈ N.
Definition 4 ([24,25]). A k-times differentiable function f (x) > 0 is said to be k-log-convex (or k-log-concave, respectively)on an interval I , with k ∈ N, if and only if [ln f (x)](k) ≥ 0 (or [ln f (x)](k) ≤ 0, respectively) on I .
The former two conclusions in Theorem 1, which were ever circulated in the preprint [26], have been employed in theproofs of [23, Theorem 1.4] and [25, Theorem 2.2 and Theorem 2.3].(1) The special cases for α = i and β = 1 of the second conclusion in Theorem 1 were made use of to procure the followingtheorem.
Theorem 3 ([23, Theorem 1.4]). The function
Gs,t(x) =[0(1+ tx)]s
[0(1+ sx)]t(2.9)
for x, s, t ∈ R such that 1+ sx > 0 and 1+ tx > 0 with s 6= t has the following properties:(a) For t > s > 0 and x ∈ (0,∞), Gs,t(x) is an increasing function and a logarithmically completely monotonic function ofsecond order in x;
(b) For t > s > 0 and x ∈(−1t , 0
), Gs,t(x) is a logarithmically completely monotonic function in x;
(c) For s < t < 0 and x ∈ (−∞, 0), Gs,t(x) is a decreasing function and a logarithmically absolutely monotonic function ofsecond order in x;
(d) For s < t < 0 and x ∈(0,− 1s
), Gs,t(x) is a logarithmically completely monotonic function in x;
(e) For s < 0 < t and x ∈(−1t , 0
), Gt,s(x) is an increasing function and a logarithmically absolutely convex function in x;
(f) For s < 0 < t and x ∈(0,− 1s
), Gt,s(x) is a decreasing function and a logarithmically absolutely convex function in x.
(2) The first two conclusions in Theorem1were hired in [25], a simplified version of the preprint [24], to derive the followingtwo theorems.
Theorem 4 ([25, Theorem 2.2]). For b > a > 0 and i ∈ N, the function [0(bx)]a
[0(ax)]bis (2i+1)-log-convex and (2i)-log-concave with
respect to x ∈ (0,∞).
Theorem 5 ([25, Theorem 2.3]). For b > a > 0, i ∈ N and β ≥ 12 , the function
[0(bx+β)]a
[0(ax+β)]bis (2i+ 1)-log-concave and (2i)-log-
convex with respect to x ∈ (0,∞).
Remark 4. The former two conclusions in Theorem 2, which were also issued in the preprint [26], have also been appliedin the proofs of [27, Theorem 1] and [28, Theorem 1].(1) The first result in Theorem 2 was utilized in [27, Theorem 1] to procure upper bounds for the ratio of two gammafunctions and the divided differences of the psi and polygamma functions as follows.
Theorem 6 ([27, Theorem 1]). For a > 0 and b > 0 with a 6= b, inequalities[0(a)0(b)
]1/(a−b)≤ eψ(I(a,b)) (2.10)
and
(−1)n[ψ (n−1)(a)− ψ (n−1)(b)
]a− b
≤ (−1)nψ (n)(I(a, b)) (2.11)
hold true, where n ∈ N and
I(a, b) =1e
(bb
aa
)1/(b−a)(2.12)
represents the identric or exponential mean.
2154 F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160
(2) The first two results in Theorem2were employed in [28, Theorem1] to acquire lower bounds for the ratio of two gammafunctions and the divided differences of the psi and polygamma functions and to refine the inequality (2.11).
Theorem 7 ([28, Theorem 1]). For a > 0 and b > 0 with a 6= b, the inequality
(−1)iψ (i)(Lα(a, b)) ≤(−1)i
b− a
∫ b
aψ (i)(u)du ≤ (−1)iψ (i)(Lβ(a, b)) (2.13)
holds if α ≤ −i− 1 and β ≥ −i, where i is a nonnegative integer and
Lp(a, b) =
[bp+1 − ap+1
(p+ 1)(b− a)
]1/p, p 6= −1, 0
b− aln b− ln a
, p = −1
I(a, b), p = 0
(2.14)
stands for the generalized logarithmic mean of order p ∈ R.
Remark 5. Recall [29,30,20] that a function f (x) is said to be subadditive on I if the inequality
f (x+ y) ≤ f (x)+ f (y) (2.15)
holds for all x, y ∈ I with x+ y ∈ I . If the inequality (2.15) is reversed, then f (x) is called superadditive on I .Some subadditive or superadditive properties of the gamma, psi and polygamma functions have been discovered as
follows.In [31], the function ψ(a+ x) is proved to be sub-multiplicative with respect to x ∈ [0,∞) if and only if a ≥ a0, where
a0 denotes the only positive real number which satisfies ψ(a0) = 1.In [30], the function [0(x)]α was proved to be subadditive on (0,∞) if and only if ln 2ln∆ ≤ α ≤ 0, where∆ = minx≥0
0(2x)0(x) .
In [18, Lemma 2.4], the function ψ(ex)was proved to be strictly concave on R.In [20, Theorem 3.1], the function ψ(a + ex) is proved to be subadditive on (−∞,∞) if and only if a ≥ c0, where c0 is
the only positive zero of ψ(x).In [32, Theorem 1], among other things, it was presented that the function ψ (k)(ex) for k ∈ N is concave (or convex,
respectively) on R if k = 2n− 2 (or k = 2n− 1, respectively) for n ∈ N.By the aid of the monotonicity of the function xα
∣∣ψ (i)(x+β)∣∣ in Theorem 1, the following subadditive and superadditive
properties of the function∣∣ψ (i)(ex)
∣∣ for i ∈ Nwere acquired recently.
Theorem 8 ([33,34]). For i ∈ N, the function∣∣ψ (i)(ex)
∣∣ is superadditive on (−∞, ln θ0) and subadditive on (ln θ0,∞), whereθ0 ∈ (0, 1) is the unique root of the equation 2
∣∣ψ (i)(θ)∣∣= ∣∣ψ (i)(θ2)
∣∣.Remark 6. The second conclusion in Theorem 2 was cited in [35, Lemma 2.4] and [36, Remark 2.3].
Remark 7. Theorem 8 and the facts mentioned in Remarks 3–6 show the potential applicability of Theorems 1 and 2convincingly.
Remark 8. In passing, we recollect the notion ‘‘logarithmically completely monotonic function’’ which is equivalent tothe logarithmically completely monotonic function of 0th order mentioned in Remark 3. A function f (x) is said to belogarithmically completely monotonic on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm ln fsatisfies
0 ≤ (−1)k[ln f (x)](k) <∞ (2.16)
for k ∈ N on I . By looking through the database MathSciNet, we find that this phrase was first used in [37], but with nowords to explicitly define it. Thereafter, it seems to have been ignored by the mathematical community. In early 2004,this terminology was recovered in [38] (the preprint of [39]) and it was immediately referenced in [40], the preprint ofthe paper [41]. A natural question that one may ask is: Whether is this notion trivial or not? In [38, Theorem 4], it wasproved that all logarithmically completely monotonic functions are also completely monotonic, but not conversely. Thisresult was formally published when revising [42]. Hereafter, this conclusion and its proofs were dug in [43–46] once andagain. Furthermore, in the paper [43], the logarithmically completely monotonic functions on (0,∞) were characterizedas the infinitely divisible completely monotonic functions studied in [47] and all Stieltjes transforms were proved to belogarithmically completely monotonic on (0,∞), where a function f (x) defined on (0,∞) is called a Stieltjes transform ifit can be of the form
F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160 2155
for some nonnegative number a and some nonnegative measureµ on [0,∞) satisfying∫∞
011+sdµ(s) <∞. For more infor-
mation, please refer to [43].It is remarked that many completely monotonic functions founded in a lot of literature such as [48,49,5], [1, Chapter XIII]
and the related references therein are actually logarithmically completely monotonic.
3. Lemmas
In order to verify Theorem 1, Theorem 2 and their corollaries in Sections 1.4 and 1.5, we need the following lemmas,in which Lemma 1 is simple but has been validated in [50–54] to be especially effectual in proving the monotonicity and(logarithmically) complete monotonicity of functions involving the gamma, psi and polygamma functions.
Lemma 1. Let f (x) be a function defined on an infinite interval I whose right end is∞. If limx→∞ f (x) = δ and f (x)−f (x+ε) > 0hold true for some given scalar ε > 0 and all x ∈ I , then f (x) > δ.
Proof. By mathematical induction, for all x ∈ I , we have
f (x) > f (x+ ε) > f (x+ 2ε) > · · · > f (x+ kε)→ δ
as k→∞. The proof of Lemma 1 is complete. �
Lemma 2 ([14–16]). The polygamma functions ψ (k)(x)may be expressed for x > 0 and k ∈ N as
ψ (k)(x) = (−1)k+1∫∞
0
tke−xt
1− e−tdt. (3.1)
For x > 0 and r > 0,
1xr=
10(r)
∫∞
0t r−1e−xtdt. (3.2)
For i ∈ N and x > 0,
ψ (i−1)(x+ 1) = ψ (i−1)(x)+(−1)i−1(i− 1)!
xi. (3.3)
Lemma 3. For k ∈ N, the double inequality
(k− 1)!xk
+k!2xk+1
< (−1)k+1ψ (k)(x) <(k− 1)!xk
+k!xk+1
(3.4)
holds on (0,∞).
Proof. In [48, Theorem 2.1] and [53, Lemma 1.3], the function ψ(x)− ln x+ αx was proved to be completely monotonic on
(0,∞), i.e.,
(−1)i[ψ(x)− ln x+
α
x
](i)≥ 0 (3.5)
for i ≥ 0, if and only if α ≥ 1, so is its negative, i.e., the inequality (3.5) is reversed, if and only if α ≤ 12 . In [55] and
[49, Theorem 2.1], the function ex0(x)xx−α was proved to be logarithmically completely monotonic on (0,∞), i.e.,
(−1)k[lnex0(x)xx−α
](k)≥ 0 (3.6)
for k ∈ N, if and only if α ≥ 1, so is its reciprocal, i.e., the inequality (3.6) is reversed, if and only if α ≤ 12 . Considering the
fact [41, p. 82] that a completely monotonic function which is non-identically zero cannot vanish at any point on (0,∞)and rearranging either (3.5) for i ∈ N or (3.6) for k ≥ 2 leads to the double inequality (3.4) immediately. �
4. Proofs of theorems and corollaries
Proof of Theorem 1. It is a standard argument to obtain that the function δ(t) is strictly decreasing from (0,∞) onto(0, 12
).
2156 F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160
Let gi,α,β(x) = xα∣∣ψ (i)(x+ β)
∣∣ on (0,∞). The direct calculation and rearrangement yieldsg ′i,α,β(x)
xα−1= α
∣∣ψ (i)(x+ β)∣∣−x∣∣ψ (i+1)(x+ β)
∣∣= (−1)i+1
[αψ (i)(x+ β)+ xψ (i+1)(x+ β)
]. (4.1)
Making use of (3.4) in (4.1) gives
limx→∞
g ′i,α,β(x)
xα−1= 0 (4.2)
for i ∈ N, α ∈ R and β ≥ 0. By virtue of formulas (3.3), (3.2) and (3.1) in sequence, a straightforward computation reveals
g ′i,α,β(x)
xα−1−g ′i,α,β(x+ 1)
(x+ 1)α−1= (−1)i+1
{α[ψ (i)(x+ β)− ψ (i)(x+ β + 1)
]+ x[ψ (i+1)(x+ β)− ψ (i+1)(x+ β + 1)
]− ψ (i+1)(x+ β + 1)
}=
i!α(x+ β)i+1
−(i+ 1)!x(x+ β)i+2
−(i+ 1)!(x+ β)i+2
+ (−1)i+2ψ (i+1)(x+ β)
= (−1)i+2ψ (i+1)(x+ β)+i!(α − i− 1)(x+ β)i+1
+(i+ 1)!(β − 1)(x+ β)i+2
=
∫∞
0
[t
1− e−t+ (β − 1)t + α − i− 1
]t ie−(x+β)tdt
,
∫∞
0hi,α,β(t)t ie−(x+β)tdt. (4.3)
For β = 0, easy differentiation shows that h′i,α,0(t) = −δ(t) < 0, and so the function hi,α,0(t) is strictly decreasing from(0,∞) onto (α − i− 1, α − i). Thus, if α ≥ i+ 1, the functions hi,α,0(t) and
g ′i,α,0(x)
xα−1−g ′i,α,0(x+ 1)
(x+ 1)α−1
are positive on (0,∞). Combining this with (4.2) and considering Lemma 1, it is deduced that the functionsg ′i,α,0(x)
xα−1and
g ′i,α,0(x) are positive on (0,∞). Hence, the function gi,α,0(x) is strictly increasing on (0,∞) for α ≥ i + 1. Similarly, forα ≤ i, the function gi,α,0(x) is strictly decreasing on (0,∞).For β > 0, it is easy to see that h′i,α,β(t) = −δ(t) + β , and h
′
i,α,β(t) is strictly increasing from (0,∞) onto(β − 1
2 , β).
Consequently, if β ≥ 12 , the function h
′
i,α,β(t) is positive and hi,α,β(t) is strictly increasing from (0,∞) onto (α − i,∞).Accordingly, if α ≥ i and β ≥ 1
2 , the function hi,α,β(t) is positive on (0,∞), that is,
g ′i,α,β(x)
xα−1−g ′i,α,β(x+ 1)
(x+ 1)α−1> 0 (4.4)
on (0,∞). Combining this with Lemma 1 results in the positivity of g ′i,α,β(x) on (0,∞). Therefore, for α ≥ i and β ≥12 , the
function gi,α,β(x) is strictly increasing on (0,∞).For 0 < β < 1
2 , since h′
i,α,β(t) is strictly increasing from (0,∞) onto(β − 1
2 , β), the function hi,α,β(t) attains its unique
minimum at some point t0 ∈ (0,∞)with δ(t0) = β . As a result, the unique minimum of hi,α,β(t) equals
δ−1(β)eδ−1(β)
eδ−1(β) − 1+ (β − 1)δ−1(β)+ α − i− 1,
where δ−1 is the inverse function of δ and is strictly decreasing from (0, 12 ) onto (0,∞). Consequently, when the inequal-ity (1.5) holds for 0 < β < 1
2 , the function hi,α,β(t) is positive on (0,∞), which means that the inequality (4.4) holds true.Accordingly, making use of the limit (4.2) and Lemma 1 again yields that the function gi,α,β(x) is strictly increasing on (0,∞)if 0 < β < 1
2 and the inequality (1.5) is valid. The sufficiency is proved.If gi,α,0(x) is strictly decreasing on (0,∞), then
xi+1−αg ′i,α,0(x) = αxi∣∣ψ (i)(x)
∣∣−xi+1∣∣ψ (i+1)(x)∣∣< 0. (4.5)
F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160 2157
Applying (3.4) in (4.5) and letting x→∞ lead to
0 ≥ limx→∞
xi+1−αg ′i,α,0(x)
≥ α limx→∞
xi[(i− 1)!xi+
i!2xi+1
]− limx→∞
xi+1[i!xi+1+(i+ 1)!xi+2
]= (i− 1)!(α − i),
which means α ≤ i.If gi,α,0(x) is strictly increasing on (0,∞), then
xi+2−αg ′i,α,0(x) = αxi+1∣∣ψ (i)(x)
∣∣−xi+2∣∣ψ (i+1)(x)∣∣> 0. (4.6)
Employing (3.3) and (3.4) in (4.6) and taking x→∞ results in
0 ≤ limx→0+
xi+2−αg ′i,α,0(x)
= limx→0+
{αxi+1
∣∣ψ (i)(x)∣∣−xi+2[∣∣ψ (i+1)(x+ 1)
∣∣+ (i+ 1)!xi+2
]}= α lim
x→0+xi+1
∣∣ψ (i)(x)∣∣−(i+ 1)! − lim
x→0+xi+2
∣∣ψ (i+1)(x+ 1)∣∣
≤ α limx→0+
xi+1[(i− 1)!xi+i!xi+1
]− (i+ 1)! − lim
x→0+xi+2
[i!
(x+ 1)i+1+
(i+ 1)!2(x+ 1)i+2
]= i!(α − i− 1),
thus, the necessary condition α ≥ i+ 1 follows.If the function gi,α,β(x) is strictly increasing on (0,∞) for β > 0, then
xi+1−αg ′i,α,β(x) = αxi∣∣ψ (i)(x+ β)
∣∣−xi+1∣∣ψ (i+1)(x+ β)∣∣> 0. (4.7)
Utilizing (3.4) in (4.7) and taking limit gives
0 ≤ limx→∞
xi+1−αg ′i,α,β(x)
≤ α limx→∞
xi[(i− 1)!(x+ β)i
+i!
(x+ β)i+1
]− limx→∞
xi+1[
i!(x+ β)i+1
+(i+ 1)!2(x+ β)i+2
]= (i− 1)!(α − i),
which is equivalent to α ≥ i. The proof of Theorem 1 is thus completed. �
Remark 9. The first two conclusions in Theorem 1 were ever proved by virtue of the convolution theorem for Laplacetransforms in [17, Lemma 1] and [18, Lemma 2.2], so we supply a new and unified proof for them here.
Proof of Corollary 1. It is well known [14–16] that Bernoulli polynomials Bk(x)may be defined by
text
et − 1=
∞∑k=0
Bk(x)tk
k!(4.8)
and that Bernoulli numbers Bk and Bernoulli polynomials Bk(x) are connected by Bk(1) = (−1)kBk(0) = (−1)kBk andB2k+1(0) = B2k+1 = 0 for k ≥ 1. Using these notations, the functions hi,α,β(t) and h′i,α,β(t)may be rewritten as
hi,α,β(t) =tet
et − 1+ (β − 1)t + α − i− 1
= α − i+(β −
12
)t +
∞∑k=2
Bk(1)tk
k!
= α − i+(β −
12
)t +
∞∑k=2
(−1)kBktk
k!
= α − i+(β −
12
)t +
∞∑k=1
(−1)k+1Bk+1tk+1
(k+ 1)!
= α − i+(β −
12
)t +
∞∑k=0
B2k+2t2k+2
(2k+ 2)!
2158 F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160
and
h′i,α,β(t) = β −12+
∞∑k=1
B2kt2k−1
(2k− 1)!.
The proof of Theorem 1 shows that
(1) h′i,α,0(t) < 0 on (0,∞);(2) hi,α,0(t) > 0 on (0,∞) if α ≥ i+ 1;(3) hi,α,0(t) < 0 on (0,∞) if 0 < α ≤ i;(4) h′i,α,β(t) > 0 on (0,∞) if β ≥
12 ;
(5) hi,α,β(t) > 0 on (0,∞) if α ≥ i and β ≥ 12 ;
(6) hi,α,β(t) > 0 on (0,∞) if 0 < β < 12 and inequality (1.5) holds true.
Based on these and by standard argument, Corollary 1 is thus proved. �
Proof of Theorem 2. If hi,α,β(t) ≷ 0 on (0,∞), then the function
±
∫∞
0hi,α,β(t)t ie−(x+β)tdt
is completely monotonic on (−β,∞), and so, by virtue of (4.3), it is derived that
±
[g ′i,α,β(x)xα−1
−g ′i,α,β(x+ 1)
(x+ 1)α−1
]is completely monotonic on (0,∞), that is,
(−1)j[g ′i,α,β(x)xα−1
−g ′i,α,β(x+ 1)
(x+ 1)α−1
](j)= (−1)j
[g ′i,α,β(x)xα−1
](j)− (−1)j
[g ′i,α,β(x+ 1)(x+ 1)α−1
](j)R 0 (4.9)
on (0,∞) for j ≥ 0. Moreover, formulas (3.4) and (4.1) imply
limx→∞
[g ′i,α,β(x)xα−1
](j)= limx→∞
(−1)j[g ′i,α,β(x)xα−1
](j)= 0. (4.10)
Combining (4.9) and (4.10) with Lemma 1 concludes that
(−1)j[g ′i,α,β(x)xα−1
](j)R 0,
that is, the function
±g ′i,α,β(x)
xα−1= ±
[α∣∣ψ (i)(x+ β)
∣∣−x∣∣ψ (i+1)(x+ β)∣∣]
is completely monotonic on (0,∞), if hi,α,β(t) ≷ 0 on (0,∞). In the proof of Theorem 1, we have demonstrated thathi,α,β(t) is positive on (0,∞) if either β = 0 and α ≥ i + 1, or β ≥ 1
2 and α ≥ i, or 0 < β < 12 and the inequality (1.5) is
satisfied, and that hi,α,β(t) is negative on (0,∞) if β = 0 and α ≤ i. As a result, the sufficient conditions for the functionα∣∣ψ (i)(x+ β)
∣∣−x∣∣ψ (i+1)(x+ β)∣∣ to be completely monotonic on (0,∞) follow.
The derivation of necessary conditions is same as in Theorem 1. The proof of Theorem 2 is complete. �
Proof of Corollary 2. It follows easily from Theorem 2 and the facts that
±
[α
x
∣∣ψ (i)(x+ β)∣∣−∣∣ψ (i+1)(x+ β)
∣∣] = ±1x
{α∣∣ψ (i)(x+ β)
∣∣−x∣∣ψ (i+1)(x+ β)∣∣},
that the function 1x is completely monotonic on (0,∞), and that the product of any finite completely monotonic functionsis also completely monotonic on the intersection of their domains. �
Acknowledgements
The authors appreciate the anonymous referees for their helpful and valuable comments on this manuscript. The firstauthor was partially supported by the China Scholarship Council.
F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160 2159
References
[1] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, Boston, London, 1993.[2] D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.[3] F. Hausdorff, Summationsmethoden und Momentfolgen I, Math. Z. 9 (1921) 74–109.[4] C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups: Theory of Positive Deinite and Related Functions, in: Graduate Texts inMathematics, vol. 100, Springer, Berlin, Heidelberg, New York, 1984.
[5] H. van Haeringen, Completely monotonic and related functions, Report 93–108, Faculty of Technical Mathematics and Informatics, Delft Universityof Technology, Delft, The Netherlands, 1993.
[6] K. Ball, Completely monotonic rational functions and Hall’s marriage theorem, J. Combin. Theory Ser. B 61 (1994) 118–124.[7] C.L. Frenzen, Error bounds for asymptotic expansions of the ratio of two gamma functions, SIAM J. Math. Anal. 18 (1987) 890–896.[8] J. Wimp, Sequence Transformations and their Applications, Academic Press, New York, 1981.[9] W.A. Day, On monotonicity of the relaxation functions of viscoelastic material, Proc. Cambridge Philos. Soc. 67 (1970) 503–508.[10] W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, Wiley, New York, 1966.[11] C. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups, in: Ergebnisse der Math., vol. 87, Springer, Berlin, 1975.[12] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, in: Lecture Notes in Statistics, vol. 76, Springer,
New York, 1992.[13] C.H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math. 10 (1974) 152–164.[14] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972
edition, A Wiley-Interscience Publication. Selected Government Publications. John Wiley & Sons, Inc., New York, 1984, National Bureau of Standards,Washington, DC.
[15] Zh.-X. Wang, D.-R. Guo, Special Functions Translated from the Chinese by D.-R. Guo and X.-J. Xia, World Scientific Publishing, Singapore, 1989.[16] Zh.-X. Wang, D.-R. Guo, Tèshu Hánshù Gàilùn (A Panorama of Special Functions), in: The Series of Advanced Physics of Peking University, Peking
University Press, Beijing, China, 2000 (in Chinese).[17] H. Alzer, Mean-value inequalities for the polygamma functions, Aequationes Math. 61 (1) (2001) 151–161.[18] H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004) 181–221.[19] F. Qi, W. Li, Two logarithmically completely monotonic functions connected with gamma function, RGMIA Res. Rep. Coll. 8 (3) (2005) 493–497. Art.
13. Available online at http://www.staff.vu.edu.au/rgmia/v8n3.asp.[20] P. Gao, A subadditive property of the digamma function, RGMIA Res. Rep. Coll. 8 (3) (2005) Art. 9. Available online at http://www.staff.vu.edu.au/
rgmia/v8n3.asp.[21] B.-N. Guo, R.-J. Chen, F. Qi, A class of completely monotonic functions involving the polygamma functions, J. Math. Anal. Approx. Theory 1 (2) (2006)
124–134.[22] F. Qi, Q. Yang, W. Li, Two logarithmically completely monotonic functions connected with gamma function, Integral Transforms Spec. Funct. 17 (7)
(2006) 539–542.[23] F. Qi, B.-N. Guo, S. Guo, Sh.-X. Chen, A function involving gamma function and having logarithmically absolute convexity, Integral Transforms Spec.
Funct. 18 (11) (2007) 837–843.[24] F. Qi, S. Guo, B.-N. Guo, A class of k-log-convex functions and their applications to some special functions, RGMIA Res. Rep. Coll. 10 (1) (2007) Art. 21.
Available online at http://www.staff.vu.edu.au/rgmia/v10n1.asp.[25] F. Qi, S. Guo, B.-N. Guo, Sh.-X. Chen, A class of k-log-convex functions and their applications to some special functions, Integral Transforms Spec. Funct.
19 (3) (2008) 195–200.[26] F. Qi, S. Guo, B.-N. Guo, Note on a class of completely monotonic functions involving the polygamma functions, RGMIA Res. Rep. Coll. 10 (1) (2007)
Art. 5. Available online at http://www.staff.vu.edu.au/rgmia/v10n1.asp.[27] F. Qi, S. Guo, Sh.-X. Chen, A new upper bound in the second Kershaw’s double inequality and its generalizations, J. Comput. Appl. Math. 220 (1–2)
(2008) 111–118. Available online at http://dx.doi.org/10.1016/j.cam.2007.07.037.[28] F. Qi, X.-A. Li, Sh.-X. Chen, Refinements, extensions and generalizations of the second Kershaw’s double inequality, Math. Inequal. Appl. 11 (3) (2008)
457–465.[29] H. Alzer, S. Koumandos, Sub- and super-additive properties of Fejér’s sine polynomial, Bull. London Math. Soc. 38 (2) (2006) 261–268.[30] H. Alzer, S. Ruscheweyh, A subadditive property of the gamma function, J. Math. Anal. Appl. 285 (2003) 564–577.[31] H. Alzer, O.G. Ruehr, A submultiplicative property of the psi function, J. Comput. Appl. Math. 101 (1999) 53–60.[32] J. Cao, D.-W. Niu, F. Qi, Convexities of some functions involving the polygamma functions, Appl. Math. E-Notes 8 (2008) 53–57.[33] F. Qi, B.-N. Guo, A note on additivity of polygamma functions. Available online at http://arxiv.org/abs/0903.0888.[34] F. Qi, B.-N. Guo, Subadditive and superadditive properties of polygamma functions, RGMIA Res. Rep. Coll. 10 (Suppl.) (2007) Art. 4. Available online at
http://www.staff.vu.edu.au/rgmia/v10(E).asp.[35] X.-M. Zhang, Y.-M. Chu, A double inequality for the gamma and psi functions, Int. J. Mod. Math. 3 (1) (2008) 67–73.[36] X.-M. Zhang, T.-Q. Xu, L.-B. Situ, Geometric convexity of a function involving gamma function and applications to inequality theory, J. Inequal. Pure
Appl. Math. 8 (1) (2007) Art. 17. Available online at http://jipam.vu.edu.au/article.php?sid=830.[37] R.D. Atanassov, U.V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C.R. Acad. Bulgare Sci. 41 (2) (1988)
21–23.[38] F. Qi, B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (1) (2004) 63–72. Art. 8.
Available online at http://www.staff.vu.edu.au/rgmia/v7n1.asp.[39] F. Qi, B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (1) (2010) (in press).[40] F. Qi, B.-N. Guo, Ch.-P. Chen, Some completelymonotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (1) (2004)
31–36. Art. 5. Available online at http://www.staff.vu.edu.au/rgmia/v7n1.asp.[41] F. Qi, B.-N. Guo, Ch.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006)
81–88.[42] F. Qi, Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004) 603–607.[43] C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (4) (2004) 433–439.[44] P. Gao, Some monotonicity properties of gamma and q-gamma functions. Available onlie at http://arxiv.org/abs/0709.1126.[45] A.Z. Grinshpan, M.E.H. Ismail, Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc.. 134 (2006)
1153–1160.[46] F. Qi, W. Li, B.-N. Guo, Generalizations of a theorem of I. Schur, RGMIA Res. Rep. Coll. 9 (3) (2006) Art. 15. Available online at http://www.staff.vu.edu.
au/rgmia/v9n3.asp.[47] R.A. Horn, On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheor. Verwandte. Geb 8 (1967) 219–230.[48] M.E.H. Ismail, M.E. Muldoon, Inequalities and monotonicity properties for gamma and q-gamma functions, in: R.V.M. Zahar (Ed.), in: Approximation
and Computation: A Festschrift in Honour of Walter Gautschi, ISNM, vol. 119, BirkhRauser, Basel, 1994, pp. 309–323.[49] M.E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978) 54–63.[50] F. Qi, A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums, ANZIAM J.
48 (4) (2007) 523–532.[51] F. Qi, A completely monotonic function involving divided differences of psi and polygamma functions and an application, RGMIA Res. Rep. Coll. 9 (4)
(2006) Art. 8. Available online at http://www.staff.vu.edu.au/rgmia/v9n4.asp.
2160 F. Qi et al. / Journal of Computational and Applied Mathematics 233 (2010) 2149–2160
[52] F. Qi, The best bounds in Kershaw’s inequality and two completely monotonic functions, RGMIA Res. Rep. Coll. 9 (4) (2006) Art. 2. Available online athttp://www.staff.vu.edu.au/rgmia/v9n4.asp.
[53] F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (7)(2007) 503–509.
[54] F. Qi, B.-N. Guo, Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications, Commun.Pure Appl. Anal. 8 (6) (2009) 1975–1989. Available online at http://dx.doi.org/10.3934/cpaa.2009.8.1975.
[55] Ch.-P. Chen, F. Qi, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (1) (2006) 405–411.