## Some logarithmically completely monotonic functions involving gamma function (2005)

Citations: | 10 - 6 self |

### BibTeX

@MISC{Qi05somelogarithmically,

author = {Feng Qi and Bai-ni Guo},

title = {Some logarithmically completely monotonic functions involving gamma function},

year = {2005}

}

### OpenURL

### Abstract

Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x

### Citations

2062 |
Handbook of mathematical functions
- Abramovitz, Stegun
- 1972
(Show Context)
Citation Context ...s called the psi or digamma function and ψ (n) (x) for n ∈ N the polygamma functions. It is well-known that the gamma function is a very important classical special function and has many applications =-=[1, 10, 20, 29]-=-. One of the reasons why the gamma function is still interesting, although nearly three centuries have elapsed after its first appearance, is that it has many applications to various areas of mathemat... |

255 |
The Laplace Transform
- Widder
- 1941
(Show Context)
Citation Context ...for x ∈ [0, ∞). This tells us that f ∈ C[[0, ∞)] if and only if it is a Laplace transform of the measure α. There have been a lot of literature about the completely monotonic functions, for examples, =-=[3, 4, 5, 22, 25, 32, 34, 35, 39, 40, 42, 58, 60, 61, 62, 63, 66]-=- and references therein. Recall also [6, 43, 47] that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (−1) n [lnf(... |

96 | Potential Theory on Locally Compact Abelian Groups - Berg, Forst - 1975 |

38 | On some inequalities for the gamma and psi function
- Alzer
- 1997
(Show Context)
Citation Context ...for x ∈ [0, ∞). This tells us that f ∈ C[[0, ∞)] if and only if it is a Laplace transform of the measure α. There have been a lot of literature about the completely monotonic functions, for examples, =-=[3, 4, 5, 22, 25, 32, 34, 35, 39, 40, 42, 58, 60, 61, 62, 63, 66]-=- and references therein. Recall also [6, 43, 47] that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (−1) n [lnf(... |

28 |
Some completely monotonic functions involving polygamma functions and an application
- Qi, Cui, et al.
(Show Context)
Citation Context ...logarithmically) completely monotonic functions involving the (11)SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS 3 gamma, psi, or polygamma functions are established by some mathematicians (see =-=[2, 3, 4, 5, 12, 13, 14, 15, 16, 17, 18, 22, 23, 26, 30, 44, 45, 46, 50, 51, 52, 63]-=- and the references therein). In this paper, using Leibniz’s Identity, the discrete and integral representations of polygamma functions and other analytic techniques, some functions such as 1 , [Γ(x +... |

26 |
An Obata-type Theorem
- Li, Wang
(Show Context)
Citation Context ...rs on I and (−1) n [lnf(x)] (n) ≥ 0 (3) for all x ∈ I and n ∈ N. For simplicity, let L[I] stand for the set of logarithmically completely monotonic functions on I. Among other things, it is proved in =-=[7, 10, 43, 47, 56]-=- that a logarithmically completely monotonic function is always completely monotonic, that is, L[I] ⊂ C[I], but not conversely, since a convex function may not be logarithmically convex (see [33, p. 7... |

18 |
Complete monotonicities of functions involving the gamma and digamma functions
- QI, GUO
- 2004
(Show Context)
Citation Context ...re α. There have been a lot of literature about the completely monotonic functions, for examples, [3, 4, 5, 22, 25, 32, 34, 35, 39, 40, 42, 58, 60, 61, 62, 63, 66] and references therein. Recall also =-=[6, 43, 47]-=- that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (−1) n [lnf(x)] (n) ≥ 0 (3) for all x ∈ I and n ∈ N. For sim... |

16 | Logarithmically completely monotonic functions relating to the gamma function
- Chen, Qi
(Show Context)
Citation Context ...logarithmically) completely monotonic functions involving the (11)SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS 3 gamma, psi, or polygamma functions are established by some mathematicians (see =-=[2, 3, 4, 5, 12, 13, 14, 15, 16, 17, 18, 22, 23, 26, 30, 44, 45, 46, 50, 51, 52, 63]-=- and the references therein). In this paper, using Leibniz’s Identity, the discrete and integral representations of polygamma functions and other analytic techniques, some functions such as 1 , [Γ(x +... |

15 | A class of completely monotonic functions
- ALZER, BERG
(Show Context)
Citation Context ...for x ∈ [0, ∞). This tells us that f ∈ C[[0, ∞)] if and only if it is a Laplace transform of the measure α. There have been a lot of literature about the completely monotonic functions, for examples, =-=[3, 4, 5, 22, 25, 32, 34, 35, 39, 40, 42, 58, 60, 61, 62, 63, 66]-=- and references therein. Recall also [6, 43, 47] that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (−1) n [lnf(... |

15 | Integral representation of some functions related to the gamma function
- BERG
(Show Context)
Citation Context ...rs on I and (−1) n [lnf(x)] (n) ≥ 0 (3) for all x ∈ I and n ∈ N. For simplicity, let L[I] stand for the set of logarithmically completely monotonic functions on I. Among other things, it is proved in =-=[7, 10, 43, 47, 56]-=- that a logarithmically completely monotonic function is always completely monotonic, that is, L[I] ⊂ C[I], but not conversely, since a convex function may not be logarithmically convex (see [33, p. 7... |

15 |
Chángyòng Bùděngshì (Applied Inequalities), 3rd ed., Shāndōng Kēxué Jìshù Chūbǎn Shè (Shandong Science and
- KUANG
- 2004
(Show Context)
Citation Context ...) n+1 α[(y + n)(y + 1) n+1 − (y + n + 1)y n+1 ] − n(y + 1) n+1 y n+1 (y + 1) n+1 1 yn+1 { [ α 1 + 1 〈 ( y y − (y + n + 1) n y + 1 ) n+1〉] } − 1 , where y = x + i > 0. In [11, p. 28], [27, p. 154] and =-=[28]-=-, Bernoulli’s inequality states that if x ≥ −1 and x ̸= 0 and if α > 1 or if α < 0 then (1 + x) α > 1 + αx.6 F. QI AND B.-N. GUO This means that which is equivalent to 1 + s + 1 t < ( 1 + 1 ) s+1 t (... |

15 |
Monotonicity results and inequalities for the gamma and incomplete gamma functions
- QI
- 1999
(Show Context)
Citation Context ... 31], among other things, the following monotonicity results were obtained: [Γ(1 + k)] 1/k < [Γ(2 + k)] 1/(k+1) , k ∈ N; [ ( Γ 1 + 1 )] x decreases with x > 0. x These are extended and generalized in =-=[36, 37, 38, 55]-=-, among other things: The function [Γ(r)] 1/(r−1) is increasing in r > 0. Clearly, Theorem 1 generalizes these results and extends them for the range of the argument. The first conclusion in Propositi... |

15 |
A complete monotonicity property of the Gamma function
- Qi, Chen
(Show Context)
Citation Context ...re α. There have been a lot of literature about the completely monotonic functions, for examples, [3, 4, 5, 22, 25, 32, 34, 35, 39, 40, 42, 58, 60, 61, 62, 63, 66] and references therein. Recall also =-=[6, 43, 47]-=- that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (−1) n [lnf(x)] (n) ≥ 0 (3) for all x ∈ I and n ∈ N. For sim... |

14 |
Logarithmic convexity of extended mean values
- Qi
- 1999
(Show Context)
Citation Context |

13 |
Completely monotonic functions associated with the gamma function and its q-analogues
- Ismail, Lorch, et al.
- 1986
(Show Context)
Citation Context |

12 |
Completely monotonic functions involving the gamma and q-gamma functions
- Grinshpan, Ismail
(Show Context)
Citation Context |

12 |
Completely monotonic functions
- Miller, Samko
- 2001
(Show Context)
Citation Context ...logarithmically) completely monotonic functions involving the (11)SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS 3 gamma, psi, or polygamma functions are established by some mathematicians (see =-=[2, 3, 4, 5, 12, 13, 14, 15, 16, 17, 18, 22, 23, 26, 30, 44, 45, 46, 50, 51, 52, 63]-=- and the references therein). In this paper, using Leibniz’s Identity, the discrete and integral representations of polygamma functions and other analytic techniques, some functions such as 1 , [Γ(x +... |

11 | Some properties of a class of logarithmically completely monotonic functions - Atanassov, Tsoukrovski - 1988 |

10 | On infinitely divisible matrices, kernels, and functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete - Horn - 1967 |

10 |
Some inequalities involving (r!) 1/r
- MINC, SATHRE
(Show Context)
Citation Context ...α ≤ 0 and β > 0, the reciprocal of the function defined by (56) belongs to L[(0, ∞)]. Proof. These follow from combining Theorem 6 with Theorem 1, Theorem 2, Theorem 3, and Theorem 4. □ Remark 10. In =-=[26, 31]-=-, among other things, the following monotonicity results were obtained: [Γ(1 + k)] 1/k < [Γ(2 + k)] 1/(k+1) , k ∈ N; [ ( Γ 1 + 1 )] x decreases with x > 0. x These are extended and generalized in [36,... |

10 |
The extended mean values: definition, properties, monotonicities, comparison, convexities, generalizations, and applications
- QI
(Show Context)
Citation Context |

10 |
The best bounds in Gautschi-Kershaw inequalities
- QI, GUO, et al.
(Show Context)
Citation Context |

8 |
Monotonicity results for gamma function
- CHEN, QI
(Show Context)
Citation Context |

8 |
Inequalities and monotonicity for the ratio of gamma functions
- GUO, QI
(Show Context)
Citation Context |

8 |
Monotonicity results for the gamma function, Atti
- KERSHAW, LAFORGIA
- 1985
(Show Context)
Citation Context |

7 | A Dictionary of - BULLEN - 1998 |

7 |
Some inequalities involving geometric mean of natural numbers and ratio of gamma functions
- QI, GUO
(Show Context)
Citation Context |

7 |
A monotonicity property of the Γ-function
- VOGT, VOIGT
(Show Context)
Citation Context |

6 | Inequalities involving gamma and psi function
- CLARK, ISMAIL
(Show Context)
Citation Context |

6 |
Completely monotonic and related functions
- Haeringen
- 1996
(Show Context)
Citation Context |

5 | Pick functions related to the gamma function
- Berg, Pedersen
(Show Context)
Citation Context ... non-elementary argument for the sufficient part of Theorem 2 was provided by an anonymous referee of this paper as follows. Looking at what is really written in [5], which builds on a technique from =-=[9]-=-, it is easy to obtain that with h(z) = ln Γ(z + 1) z ∫ ∞ − α ln(z + 1) = c + c = −γ + h ′ ∫ ∞ (t) = − 1 ∞∑ ( 1 k k=1 1 ( 1 t + z − ) 1 − arctan k t 1 + t2 ) [α − M(t)] dt and M(t) = k−1 t for t ∈ (k ... |

5 |
Sur un theoreme de M. S. Bernstein relatif ala transformation de Laplace-Stieltjes
- DUBOURDIEU
(Show Context)
Citation Context .... On [8, p. 127] it is proved that S ⊂ H. From Bernstein’s Theorem it also follows that completely monotonic functions on (0, ∞) are always strictly completely monotonic unless they are constant, see =-=[19, 53]-=- and [61, p. 11]. Also it follows that a logarithmically completely monotonic function on (0, ∞) is strictly so unless it is of the form c exp(−αx) for c > 0 and α ≥ 0, so there is no need to discuss ... |

5 |
Sur la fonction Gamma, Publ
- SÁNDOR
- 1989
(Show Context)
Citation Context ...m and n. Remark 11. The results in Proposition 1 generalize and extend those of [48, 49]. Define Qa,b(x) = [Γ(x + a + 1)]1/(x+a) [Γ(x + b + 1)] 1/(x+b) for nonnegative real numbers a and b. J. Sándor =-=[59]-=- established that Q1,0 is decreasing on (1, ∞). In [5] Alzer and Berg proved that [Qa,b(x)] c is completely monotonic with x ∈ (0, ∞) if and only if a ≥ b for c > 0. The following proposition extends ... |

4 |
Inequalities for the Gamma function relating to asymptotic expansions
- Allasia, Giordano, et al.
- 2002
(Show Context)
Citation Context |

4 |
Harmonic Analysis and the Theory of Probability, California Monographs
- Bochner
- 1960
(Show Context)
Citation Context ...rs on I and (−1) n [lnf(x)] (n) ≥ 0 (3) for all x ∈ I and n ∈ N. For simplicity, let L[I] stand for the set of logarithmically completely monotonic functions on I. Among other things, it is proved in =-=[7, 10, 43, 47, 56]-=- that a logarithmically completely monotonic function is always completely monotonic, that is, L[I] ⊂ C[I], but not conversely, since a convex function may not be logarithmically convex (see [33, p. 7... |

4 |
Logarithmic convexities of the extended mean values, RGMIA Res
- Qi
- 1999
(Show Context)
Citation Context |

3 |
Completely monotonic function associated with the gamma function and proof of Wallis’ inequality
- Chen, Qi
(Show Context)
Citation Context |

3 |
Inequalities relating to the gamma function
- Chen, Qi
(Show Context)
Citation Context |

3 | Chángyòng Bùděngshì (Applied - KUANG - 1993 |

3 |
Special Functions, Translated from the Chinese by Guo and X
- Wang, Guo
- 1989
(Show Context)
Citation Context ...hese theorems. In Section 3, some remarks are given, some new results are deduced, and some known results are recovered, as applications of these theorems. 2. Proofs of theorems It is well-known (see =-=[1, 20, 64, 65]-=- and [29, p. 16]) that the polygamma functions ψ (k)(x) can be expressed for x > 0 and k ∈ N as ψ (k) (x) = (−1) k+1 ∞∑ 1 k! (x + i) k+1 (22) or i=0 ψ (k) (x) = (−1) k+1 ∫ ∞ The first proof of Theorem... |

2 |
Logarithmically completely monotonic ratios of mean values and an application
- Chen, Qi
(Show Context)
Citation Context |

2 |
Monotonicity and convexity results for functions involving the gamma function
- Qi, Chen
(Show Context)
Citation Context |

2 |
On a new generalization of Martins’ inequality
- Qi, Guo
(Show Context)
Citation Context ... 31], among other things, the following monotonicity results were obtained: [Γ(1 + k)] 1/k < [Γ(2 + k)] 1/(k+1) , k ∈ N; [ ( Γ 1 + 1 )] x decreases with x > 0. x These are extended and generalized in =-=[36, 37, 38, 55]-=-, among other things: The function [Γ(r)] 1/(r−1) is increasing in r > 0. Clearly, Theorem 1 generalizes these results and extends them for the range of the argument. The first conclusion in Propositi... |

2 |
The function (bx −ax )/x: Inequalities and properties
- Qi, Xu
- 1998
(Show Context)
Citation Context |

2 |
Tèshū Hánshù Gàilùn (A Panorama of Special Functions), The Series of Advanced
- Wang, Guo
- 2000
(Show Context)
Citation Context ...hese theorems. In Section 3, some remarks are given, some new results are deduced, and some known results are recovered, as applications of these theorems. 2. Proofs of theorems It is well-known (see =-=[1, 20, 64, 65]-=- and [29, p. 16]) that the polygamma functions ψ (k)(x) can be expressed for x > 0 and k ∈ N as ψ (k) (x) = (−1) k+1 ∞∑ 1 k! (x + i) k+1 (22) or i=0 ψ (k) (x) = (−1) k+1 ∫ ∞ The first proof of Theorem... |

1 |
On a new generalization of Martins’ inequality, RGMIA Res
- Qi
(Show Context)
Citation Context ... 31], among other things, the following monotonicity results were obtained: [Γ(1 + k)] 1/k < [Γ(2 + k)] 1/(k+1) , k ∈ N; [ ( Γ 1 + 1 )] x decreases with x > 0. x These are extended and generalized in =-=[36, 37, 38, 55]-=-, among other things: The function [Γ(r)] 1/(r−1) is increasing in r > 0. Clearly, Theorem 1 generalizes these results and extends them for the range of the argument. The first conclusion in Propositi... |

1 |
An infimum and an upper bound of a function with two independent variables
- Qi, Niu, et al.
(Show Context)
Citation Context ...)] = 1. (50) Since µ ∈ (0, ∞) and t ∈ (0, ∞) are arbitrary, so we have τ(s, t) < 1 for (s, t) ∈ (0, ∞) × (0, ∞). Recently, the upper bound of τ(s, t) was improved from 1 to 1 3 in [57] and further in =-=[41]-=-. to 3 10 Remark 8. By definition, it is clear that one of the necessary conditions such that (x+1) α [Γ(x+1)] 1/x ∈ L[(−1, ∞)] is [lnνα(x)] ′ ≥ 0 in (−1, ∞), where να(x) is defined by (29), which is ... |

1 |
A complete monotonicity of the gamma function
- Qi, Chen
(Show Context)
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1 |
Monotonicity and convexity of the function x+α
- Qi, Guo
(Show Context)
Citation Context ...analysis, number theory, potential theory, probability theory [10], physics [29], numerical and asymptotic analysis, integral transforms [66], and combinatorics. Some related references are listed in =-=[3, 4, 5, 7, 22, 32, 48, 49, 53, 66]-=-. In recent years, inequalities and (logarithmically) completely monotonic functions involving the (11)SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS 3 gamma, psi, or polygamma functions are est... |

1 |
Monotonicity and convexity of ratio between gamma functions to different powers
- Qi, Guo
(Show Context)
Citation Context ...analysis, number theory, potential theory, probability theory [10], physics [29], numerical and asymptotic analysis, integral transforms [66], and combinatorics. Some related references are listed in =-=[3, 4, 5, 7, 22, 32, 48, 49, 53, 66]-=-. In recent years, inequalities and (logarithmically) completely monotonic functions involving the (11)SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS 3 gamma, psi, or polygamma functions are est... |