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Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic

Journal of Inequalities and Applications20112011:36

https://doi.org/10.1186/1029-242X-2011-36

©  Lv et al; licensee Springer. 2011

  • Received: 9 February 2011
  • Accepted: 22 August 2011
  • Published:

Abstract

We prove that the function F α,β (x) = x α Γ β (x)/Γ(βx) is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) ∈ {(α, β) : β > 0, β ≥ 2α + 1, β ≥ α + 1}\{(α, β) : α = 0, β = 1} and that [F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) ∈ {(α, β ) : β > 0, β ≤ 2α + 1, β ≤ α + 1}\{(α, β ) : α = 0, β = 1}.

2010 Mathematics Subject Classification: 33B15; 26A48.

Keywords

  • completely monotonic
  • logarithmically completely monotonic
  • gamma function

1 Introduction

For real and positive values of x the Euler gamma function Γ and its logarithmic derivative ψ, the so-called digamma functions are defined by
Γ ( x ) = ∫ 0 ∞ t x - 1 e - t d t ,
(1.1)
ψ ( x ) = Γ ′ ( x ) Γ ( x ) = - γ + ∫ 0 ∞ e - t - e - x t 1 - e - t d t ,
(1.2)

where γ = 0.5772 ··· is the Euler's constant.

For extension of these functions to complex variable and for basic properties see [1]. Over the last half century, many authors have established inequalities and monotonicity for these functions [2–22].

We know that a real-valued function f : I → ℝ is said to be completely monotonic on I if f has derivatives of all orders on I and
( - 1 ) n f ( n ) ( x ) ≥ 0
(1.3)

for all x ∈ I and n ≥ 0. Moreover, f is said to be strictly completely monotonic if inequalities (1.3) are strict.

We also know that a positive real-valued function f : I → (0, ∞) is said to be logarithmically completely monotonic on I if f has derivatives of all orders on I and its logarithm log f satisfies
( - 1 ) k [ l o g f ( x ) ] ( k ) ≥ 0
(1.4)

for all x ∈ I and k ∈ ℕ. Moreover, f is said to be strictly logarithmically completely monotonic if inequalities (1.4) are strict.

Recently, the completely monotonic or logarithmically completely monotonic functions have been the subject of intensive research. In particular, many complete monotonicity and logarithmically complete monotonicity properties related to the gamma function, psi function, and polygamma function can be found in the literature [17, 18, 23–37]. In 1997, Merkle [38] proved that F ( x ) = Γ ( 2 x ) Γ 2 ( x ) is strictly log-concave on (0, ∞). Later, Chen [39] showed that [ F ( x ) ] - 1 = Γ 2 ( x ) Γ ( 2 x ) is strictly logarithmically completely monotonic on (0, ∞). In [40], Li and Chen proved that F β ( x ) = Γ β ( x ) Γ ( β x ) is strictly logarithmically completely monotonic on (0, ∞) for β > 1, and that [F β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞) for 0 < β < 1. The purpose of this article is to generalize Li and Chen's result. Our main result is as follows.

Theorem 1.1 Let α ∈ ℝ, β > 0 and F α,β (x) = x α Γ β (x)/Γ(βx), then
  1. (1)

    F α,β (x) is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) ∈ {(α, β) : β > 0, β ≥ 2α + 1, β ≥ α + 1}\{(α, β) : α = 0, β = 1};

     
  2. (2)

    [F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) ∈ {(α, β) : β > 0, β ≤ 2α + 1, β ≤ α + 1}\{(α, β) : α = 0, β = 1}.

     

2 Lemma

In order to prove our Theorem 1.1, we need a lemma which we present in this section.

Lemma 2.1 Let α ∈ ℝ, β ∈ (0, 1) ∪ (1, ∞) and
h ( t ) = - α e - ( β + 1 ) t + ( α + 1 ) e - β t + ( α - β ) e - t + β - α - 1 .
Then the following statements are true:
  1. (1)

    If β ≤ α + 1 and β ≤ 2α + 1, then h(t) < 0 for t ∈ (0, ∞);

     
  2. (2)

    If α + 1 < β < 2α + 1, then there exists λ 1 ∈ (0, ∞) such that h(t) < 0 for t ∈ (0, λ 1) and h(t) > 0 for t ∈ (λ 1, ∞);

     
  3. (3)

    If β ≥ α + 1 and β ≥ 2α + 1, then h(t) > 0 for t ∈ (0, ∞);

     
  4. (4)

    If 2α + 1 < β < α + 1, then there exists λ 2 ∈ (0, ∞) such that h(t) > 0 for t ∈ (0, λ 2) and h(t) < 0 for t ∈ (λ 2, ∞).

     
Proof Let h1(t) = e(β + 1)th'(t) and h 2 ( t ) = e - t h 1 ′ ( t ) . Then simple computations lead to
h ( 0 ) = 0 ,
(2.1)
h ′ ( t ) = α ( β + 1 ) e - ( β + 1 ) t - β ( α + 1 ) e - β t - ( α - β ) e - t , h 1 ( 0 ) = h ′ ( 0 ) = 0 ,
(2.2)
h 1 ( t ) = α ( β + 1 ) - β ( α + 1 ) e t - ( α - β ) e β t , h ′ 1 ( t ) = - β ( α + 1 ) e t - β ( α - β ) e β t ,
(2.3)
h 2 ( 0 ) = h 1 ′ ( 0 ) = β ( β - 2 α - 1 ) ,
(2.4)
h 2 ( t ) = - β ( α + 1 ) - β ( α - β ) e ( β - 1 ) t
(2.5)
and
h 2 ′ ( t ) = β ( β - 1 ) ( β - α ) e ( β - 1 ) t .
(2.6)
  1. (1)

    If β ≤ α + 1 and β ≤ 2α + 1, then we divide the proof into four cases.

     
Case 1 If 0 < β < 1 and α < β ≤ 2α + 1, then from (2.4) and (2.6) we clearly see that
h 2 ( 0 ) ≤ 0 ,
(2.7)
h 2 ′ ( t ) < 0 .
(2.8)

Therefore, h(t) < 0 for t ∈ (0, ∞), which follows from (2.7) and (2.8) together with (2.1) and (2.2).

Case 2 If 0 < β < 1 and β ≤ α, then (2.5) and (2.6) lead to
lim t → + ∞ h 2 ( t ) = - β ( α + 1 ) < 0 ,
(2.9)
h 2 ′ ( t ) ≥ 0 .
(2.10)

Therefore, h(t) < 0 for t ∈ (0, ∞), which follows from (2.9) and (2.10) together with (2.1) and (2.2).

Case 3 If 1 < β ≤ α, then (2.4) and (2.6) lead to
h 2 ( 0 ) < 0 ,
(2.11)
h 2 ′ ( t ) ≤ 0 .
(2.12)

From equations (2.1) and (2.2) together with inequalities (2.11) and (2.12), we clearly see that h(t) < 0 for t ∈ (0, ∞).

Case 4 If β > 1 and α < β ≤ α + 1, then we clearly see that
lim t → + ∞ h ( t ) = β - α - 1 ≤ 0 .
(2.13)
From (2.3)-(2.6), we know that
lim t → + ∞ h 1 ( t ) = + ∞ ,
(2.14)
h 2 ( 0 ) < 0 ,
(2.15)
lim t → + ∞ h 2 ( t ) = + ∞ ,
(2.16)
h 2 ′ ( t ) > 0 .
(2.17)

From (2.15)-(2.17), we clearly see that there exists t1> 0 such that h2(t) < 0 for t ∈ (0, t1) and h2(t) > 0 for t ∈ (t1, ∞). Hence, h1(t) is strictly decreasing in [0, t1] and strictly increasing in [t1, ∞).

From (2.2) and (2.14) together with the monotonicity of h1(t), we know that there exists t2> 0 such that h1(t) < 0 for t ∈ (0, t2) and h1(t) > 0 for t ∈ (t2, ∞). Hence, h(t) is strictly decreasing in [0, t2] and strictly increasing in [t2, ∞).

Therefore, h(t) < 0 for t ∈ (0, ∞) follows from (2.1) and (2.13) together with the monotonicity of h(t).

(2) If α + 1 < β < 2α + 1, then we clearly see that
lim t → + ∞ h ( t ) = β - α - 1 > 0
(2.18)

and (2.14)-(2.17) hold again. From the proof of Case 4 in Lemma 2.1(1), we know that there exists λ > 0 such that h(t) is strictly decreasing in [0, λ] and strictly increasing in [λ, ∞).

Therefore, Lemma 2.1(2) follows from (2.1) and (2.18) together with the monotonicity of h(t).

(3) If β ≥ α + 1 and β ≥ 2α + 1, then we divide the proof into three cases.

Case I If β > 1 and β ≥ 2α + 1, then
β > α
(2.19)
and it follows from (2.4) that
h 2 ( 0 ) ≥ 0 .
(2.20)
Equation (2.6) and inequality (2.19) lead to
h 2 ′ ( t ) > 0 .
(2.21)

Therefore, h(t) > 0 for t ∈ (0, ∞) follows from (2.1), (2.2), (2.20), and (2.21).

Case II If 0 < β < 1 and α ≤ -1, then from (2.5) and (2.6) we clearly see that
lim t → + ∞ h 2 ( t ) = - β ( α + 1 ) ≥ 0 ,
(2.22)
h 2 ′ ( t ) < 0 .
(2.23)
Inequalities (2.22) and (2.23) imply that
h 2 ( t ) > 0
(2.24)

for t ∈ (0, ∞).

Therefore, h(t) > 0 for t ∈ (0, ∞) follows from (2.1) and (2.2) together with (2.24).

Case III If 0 < α + 1 ≤ β < 1, then we clearly see that
lim t → + ∞ h ( t ) = β - α - 1 ≥ 0 .
(2.25)
It follows from (2.3)-(2.6) that
lim t → + ∞ h 1 ( t ) = - ∞ ,
(2.26)
h 2 ( 0 ) = β ( β - 2 α - 1 ) > 0 ,
(2.27)
lim t → + ∞ h 2 ( t ) = - β ( α + 1 ) < 0 ,
(2.28)
h 2 ′ ( t ) < 0 .
(2.29)

Inequalities (2.27)-(2.29) imply that there exists t3> 0 such that h2(t) > 0 for t ∈ (0, t3) and h2(t) < 0 for t ∈ (t3, ∞). Hence, h1(t) is strictly increasing in [0, t3] and strictly decreasing in [t3, ∞).

It follows from (2.2) and (2.26) together with the monotonicity of h1(t) that there exists t4> 0 such that h1(t) > 0 for t ∈ (0, t4) and h1(t) < 0 for t ∈ (t4, ∞). Hence, h(t) is strictly increasing in [0, t4] and strictly decreasing in [t4, ∞).

Therefore, h(t) > 0 for t ∈ (0, ∞) follows from (2.1) and (2.25) together with the monotonicity of h(t).

(4) If 2α + 1 < β < α + 1, then we clearly see that
lim t → + ∞ h ( t ) = β - α - 1 < 0
(2.30)

and (2.26)-(2.29) hold again.

From the proof of Case III in Lemma 2.1(3) we know that there exists μ > 0 such that h(t) is strictly increasing in [0, μ] and strictly decreasing in [μ, ∞).

Therefore, Lemma 2.1(4) follows from (2.1) and (2.30) together with the monotonicity of h(t).

3 Proof of Theorem 1.1

Proof of Theorem 1.1 Let E1 = {(α, β) : 0 < β < 1, β ≥ α + 1}, E2 = {(α, β) : β > 1, β ≥ 2α +1}, E3 = {(α, β) : α < 0, β = 1}, E4 = {(α, β) : α = 0, β = 1}, E5 = {(α, β) : α + 1 < β < 2α + 1}, E6 = {(α, β) : β > 0, 2α + 1 < β < α + 1}, E7 = {(α, β) : 0 < β < 1, β ≤ 2α + 1}, E8 = {(α, β) : β > 1, β ≤ α + 1} and E9 = {(α, β) : α > 0, β = 1}. Then
{ ( α , β ) : α ∈ ℝ , β > 0 } = ∪ i = 1 9 E i , { ( α , β ) : β > 0 , β ≥ 2 α + 1 , β ≥ α + 1 } \ { ( α , β ) : α = 0 , β = 1 } = E 1 ∪ E 2 ∪ E 3 , { ( α , β ) : β > 0 , β ≤ 2 α + 1 , β ≤ α + 1 } \ { ( α , β ) : α = 0 , β = 1 } = E 7 ∪ E 8 ∪ E 9 .
  1. (1)

    We divide the proof of Theorem 1.1(1) into five cases.

     
Case 1.1 (α, β) ∈ E1 ∪ E2. From (1.1), (1.2), and applying
ψ m ( x ) = ( − 1 ) m + 1 ∫ 0 ∞ t m 1 − e − t e − x t d t ( x > 0, m = 1,2,... ),
we obtain for n ≥ 1,
( − 1 ) n [ log F α , β ( x ) ] ( n ) = ( − 1 ) n [ ( − 1 ) n − 1 α ( n − 1 ) ! x n + β ψ ( n − 1 ) ( x ) − β n ψ ( n − 1 ) ( β x ) ] = − α ∫ 0 ∞ s n − 1 e − x s d s + β ∫ 0 ∞ s n − 1 1 − e − s e − x s d s − β n ∫ 0 ∞ t n − 1 1 − e − t e − β x t d t = − α β n ∫ 0 ∞ t n − 1 e − β x t d t + β n + 1 ∫ 0 ∞ t n − 1 1 − e − β t e − β x t d t − β n ∫ 0 ∞ t n − 1 1 − e − t e − β x t d t = β n ∫ 0 ∞ t n − 1 e − β x t ( 1 − e − t ) ( 1 − e − β t ) h ( t ) d t ,
(3.1)
where
h ( t ) = - α e - ( β + 1 ) t + ( α + 1 ) e - β t + ( α - β ) e - t - α + β - 1 .
(3.2)

Therefore, F α,β (x) is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.1) and (3.2) together with Lemma 2.1(3).

Case 1.2 (α, β) ∈ E3. Then we clearly see that
( − 1 ) n [ l o g F α , β ( x ) ] ( n ) = ( − 1 ) n α ( n − 1 ) ! ( − 1 ) n − 1 x n = − α ( n − 1 ) ! x n > 0
(3.3)

for all x > 0.

Therefore, F α,β (x) is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.3).

Case 1.3 (α,β) ∈ E4. Then F α,β (x) = 1 and
( - 1 ) n [ l o g F α , β ( x ) ] ( n ) = 0 .
(3.4)

Therefore, F α,β (x) is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.4).

Case 1.4 (α, β) ∈ E5∪E6∪E7∪E8. Then F α,β (x) is not strictly logarithmically completely monotonic on (0, ∞), which follows from Lemmas 2.1(2), 2.1(4), 2.1(1), and equations (3.1) and (3.2).

Case 1.5 (α, β) ∈ E9. Then
( - 1 ) n [ l o g F α , β ( x ) ] ( n ) = - α ( n - 1 ) ! x n < 0
(3.5)

for all x > 0.

Therefore, F α,β (x) is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.5).
  1. (2)

    We divide the proof of Theorem 1.1(2) into five cases.

     
Case 2.1 (α, β) ∈ E7 ∪ E8. Then from (3.1) we get
( - 1 ) n { l o g [ F α , β ( x ) ] - 1 } ( n ) = - β n ∫ 0 ∞ t n - 1 e - β x t ( 1 - e - t ) ( 1 - e - β t ) h ( t ) d t ,
(3.6)
where
h ( t ) = - α e - ( β + 1 ) t + ( α + 1 ) e - β t + ( α - β ) e - t - α + β - 1 .
(3.7)

Therefore, [F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.6) and (3.7) together with Lemma 2.1(1).

Case 2.2 (α, β) ∈ E9. Then
( - 1 ) n { l o g [ F α , β ( x ) ] - 1 } ( n ) = α ( n - 1 ) ! x n > 0
(3.8)

for all x > 0.

Therefore, [F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.8).

Case 2.3 (α, β) ∈ E4. Then [F α,β (x)]-1 = 1 and
( - 1 ) n { l o g [ F α , β ( x ) ] - 1 } ( n ) = 0 .
(3.9)

Therefore, [F α,β (x)]-1 is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.9).

Case 2.4 (α, β) ∈ E1 ∪ E2 ∪ E5 ∪ E6. Then [F α,β (x)]-1 is not strictly logarithmically completely monotonic on (0, ∞), which follows from equations (3.6) and (3.7) and Lemmas 2.1(3), 2.1(2) and 2.1(4).

Case 2.5 (α, β) ∈ E3. Then
( - 1 ) n { l o g [ F α , β ( x ) ] - 1 } ( n ) = α ( n - 1 ) ! x n < 0
(3.10)

for all x > 0.

Therefore, [F α,β (x)]-1 is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.10). □

Declarations

Acknowledgements

This study is partly supported by the Natural Science Foundation of China (Grant no. 11071069), the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant no. T200924), and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant no. Y200805602).

Authors’ Affiliations

(1)
Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, PR China

References

  1. Whittaker ET, Watson GN: A Course of Modern Analysis. Cambridge University Press, New York; 1962.Google Scholar
  2. Chu Y-M, Zhang X-M, Zhang Zh-H: The geometric convexity of a function involving gamma function with applications. Commun Korean Math Soc 2010,25(3):373–383. 10.4134/CKMS.2010.25.3.373MathSciNetView ArticleGoogle Scholar
  3. Zhang X-M, Chu Y-M: A double inequality for gamma function. J Inequal Appl 2009, 2009: 1–7. Article ID 503782MathSciNetGoogle Scholar
  4. Zhao T-H, Chu Y-M, Jiang Y-P: Monotonic and logarithmically convex properties of a function involving gamma functions. J Inequal Appl 2009, 2009: 1–13. Article ID 728612MathSciNetGoogle Scholar
  5. Zhang X-M, Chu Y-M: An inequality involving the gamma function and the psi function. Int J Mod Math 2008,3(1):67–73.MathSciNetGoogle Scholar
  6. Chu Y-M, Zhang X-M, Tang X-M: An elementary inequality for psi function. Bull Inst Math Acad Sin 2008,3(3):373–380.MathSciNetGoogle Scholar
  7. Song Y-Q, Chu Y-M, Wu L-L: An elementary double inequality for gamma function. Int J Pure Math 2007,38(4):549–554.MathSciNetGoogle Scholar
  8. Guo B-N, Qi F: Two new proofs of the complete monotonicity of a function involving the PSI function. Bull Korean Math Soc 2010,47(1):103–111. 10.4134/BKMS.2010.47.1.103MathSciNetView ArticleGoogle Scholar
  9. Chen Ch-P, Qi F, Srivastava HM: Some properties of functions related to the gamma and psi functions. Int Trans Spec Funct 2010,21(1–2):153–164.MathSciNetView ArticleGoogle Scholar
  10. Qi F: A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums. ANZIAM J 2007,48(4):523–532. 10.1017/S1446181100003199MathSciNetView ArticleGoogle Scholar
  11. Qi F, Guo B-N: Monotonicity and convexity of ratio between gamma functions to different powers. J Indones Math Soc 2005,11(1):39–49.MathSciNetGoogle Scholar
  12. Chen Ch-P, Qi F: Inequalities relating to the gamma function. Aust J Math Anal Appl 2004,1(1):1–7. Article 3MathSciNetGoogle Scholar
  13. Guo B-N, Qi F: Inequalities and monotonicity for the ratio of gamma functions. Taiwan J Math 2003,7(2):239–247.MathSciNetGoogle Scholar
  14. Qi F: Monotonicity results and inequalities for the gamma and incomplete gamma functions. Math Inequal Appl 2002,5(1):61–67.MathSciNetGoogle Scholar
  15. Qi F, Mei J-Q: Some inequalities of the incomplete gamma and related functions. Z Anal Anwendungen 1999,18(3):793–799.MathSciNetView ArticleGoogle Scholar
  16. Qi F, Guo S-L: Inequalities for the incomplete gamma and related functions. Math Inequal Appl 1999,2(1):47–53.MathSciNetGoogle Scholar
  17. Alzer H: Some gamma function inequalities. Math Comp 1993,60(201):337–346. 10.1090/S0025-5718-1993-1149288-7MathSciNetView ArticleGoogle Scholar
  18. Alzer H: On some inequalities for the gamma and psi functions. Math Comp 1997,66(217):373–389. 10.1090/S0025-5718-97-00807-7MathSciNetView ArticleGoogle Scholar
  19. Anderson GD, Qiu S-L: A monotonicity property of the gamma function. Proc Am Math Soc 1997,125(11):3355–3362. 10.1090/S0002-9939-97-04152-XMathSciNetView ArticleGoogle Scholar
  20. Kershaw D: Some extensions of W. Gautschi's inequalities for the gamma function. Math Comp 1983,41(164):607–611.MathSciNetGoogle Scholar
  21. Merkle M: Logarithmic convexity and inequalities for the gamma function. J Math Anal Appl 1996,203(2):369–380. 10.1006/jmaa.1996.0385MathSciNetView ArticleGoogle Scholar
  22. Palumbo B: A generalization of some inequalities for the gamma function. J Comput Appl Math 1998,88(2):255–268. 10.1016/S0377-0427(97)00187-8MathSciNetView ArticleGoogle Scholar
  23. Alzer H, Berg Ch: Some classes of completely monotonic functions II. Ramanujan J 2006,11(2):225–248. 10.1007/s11139-006-6510-5MathSciNetView ArticleGoogle Scholar
  24. Alzer H: Sharp inequalities for the digamma and polygamma functions. Forum Math 2004,16(2):181–221. 10.1515/form.2004.009MathSciNetView ArticleGoogle Scholar
  25. Alzer H, Batir N: Monotonicity properties of the gamma function. Appl Math Lett 2007,20(7):778–781. 10.1016/j.aml.2006.08.026MathSciNetView ArticleGoogle Scholar
  26. Clark WE, Ismail MEH: Inequalities involving gamma and psi functions. Anal Appl 2003,1(1):129–140. 10.1142/S0219530503000041MathSciNetView ArticleGoogle Scholar
  27. Elbert Á, Laforgia A: On some properties of the gamma function. Proc Am Math Soc 2000,128(9):2267–2673.MathSciNetView ArticleGoogle Scholar
  28. Bustoz J, Ismail MEH: On gamma function inequalities. Math Comp 1986,47(176):659–667. 10.1090/S0025-5718-1986-0856710-6MathSciNetView ArticleGoogle Scholar
  29. Ismail MEH, Lorch L, Muldoon ME: Completely monotonic functions associated with the gamma function and its q-analogues. J Math Anal Appl 1986,116(1):1–9. 10.1016/0022-247X(86)90042-9MathSciNetView ArticleGoogle Scholar
  30. Babenko VF, Skorokhodov DS: On Kolmogorov-type inequalities for functions defined on a semiaxis. Ukr Math J 2007,59(10):1299–1312.MathSciNetView ArticleGoogle Scholar
  31. Muldoon ME: Some monotonicity properties and characterizations of the gamma function. Aequationes Math 1978,18(1–2):54–63. 10.1007/BF01844067MathSciNetView ArticleGoogle Scholar
  32. Qi F, Yang Q, Li W: Two logarithmically completely monotonic functions connected with gamma function. Int Trans Spec Funct 2006,17(7):539–542. 10.1080/10652460500422379MathSciNetView ArticleGoogle Scholar
  33. Qi F, Niu D-W, Cao J: Logarithmically completely monotonic functions involving gamma and polygamma functions. J Math Anal Approx Theory 2006,1(1):66–74.MathSciNetGoogle Scholar
  34. Qi F, Chen Sh-X, Cheung W-S: Logarithmically completely monotonic functions concerning gamma and digamma functions. Int Trans Spec Funct 2007,18(6):435–443. 10.1080/10652460701318418MathSciNetView ArticleGoogle Scholar
  35. Qi F: A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality. J Comput Appl Math 2007,206(2):1007–1014. 10.1016/j.cam.2006.09.005MathSciNetView ArticleGoogle Scholar
  36. Chen Ch-P, Qi F: Logarithmically complete monotonicity properties for the gamma function. Aust J Math Anal Appl 2005,2(2):1–9. Article 8MathSciNetGoogle Scholar
  37. Chen Ch-P, Qi F: Logarithmically completely monotonic functions relating to the gamma function. J Math Anal Appl 2006,321(1):405–411. 10.1016/j.jmaa.2005.08.056MathSciNetView ArticleGoogle Scholar
  38. Merkle M: On log-convexity of a ratio of gamma functions. Univ Beograd Publ Elektrotehn Fak Ser Mat 1997, 8: 114–119.MathSciNetGoogle Scholar
  39. Chen Ch-P: Complete monotonicity properties for a ratio of gamma functions. Univ Beograd Publ Elektrotehn Fak Ser Mat 2005, 16: 26–28.MathSciNetGoogle Scholar
  40. Li A-J, Chen Ch-P: Some completely monotonic functions involving the gamma and polygamma functions. J Korean Math Soc 2008,45(1):273–287. 10.4134/JKMS.2008.45.1.273MathSciNetView ArticleGoogle Scholar

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© Lv et al; licensee Springer. 2011

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.

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