## 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

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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... |

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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(... |

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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(... |

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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 +... |

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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... |

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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
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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
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(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
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(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
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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 (... |

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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
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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... |

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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 +... |

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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,... |

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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 ... |

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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 ... |

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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 ... |

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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 |
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3 | Chángyòng Bùděngshì (Applied - KUANG - 1993 |

3 |
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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
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2 |
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On a new generalization of Martins’ inequality
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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 |
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2 |
Tèshū Hánshù Gàilùn (A Panorama of Special Functions), The Series of Advanced
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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
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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
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1 |
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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... |