## TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION (902)

### BibTeX

@MISC{Qi902twonew,

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

title = {TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION},

year = {902}

}

### OpenURL

### Abstract

Abstract. In the present paper, we give two new proofs for the necessary and sufficient condition α ≤ 1 such that the function x α [lnx − ψ(x)] is completely monotonic on (0, ∞). 1.

### Citations

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Citation Context ... two limits and, for α < 1, lim x→0 + θ1(x) = 1, lim x→0 + θα(x) = ∞, 2. Lemmas lim x→∞ θ1(x) = 1 2 (16) lim x→∞ θα(x) = 0. (17) In order to prove Theorem 1, the following lemmas are needed. Lemma 1 (=-=[1]-=-). For i ∈ N, x > 0, a > 0 and b > 0, ψ (i−1) (x + 1) = ψ (i−1) (x) + (−1)i−1 (i − 1)! ln b a = ∫ ∞ 0 ψ (i) (x) = (−1) i+1 ψ(x) − lnx + 1 x = Lemma 2 ([16]). For x > 0, Lemma 3. Inequalities and (k − ... |

253 |
The Laplace Transform
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Citation Context ...stract. In the present paper, we give two new proofs for the necessary and sufficient condition α ≤ 1 such that the function x α [lnx − ψ(x)] is completely monotonic on (0, ∞). 1. Introduction Recall =-=[19]-=- that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and (−1) n f (n) (x) ≥ 0 (1) for all x ∈ I and n ∈ N ∪ {0}. The well-known Bernstein’s Th... |

38 | On some inequalities for the gamma and psi function - Alzer - 1997 |

28 |
Some completely monotonic functions involving polygamma functions and an application
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Citation Context ...τ 120x 3 1 1 < lnx − ψ(x) < (7) 2x x for x > 0 is concluded. This extends a result in [12], which says that the inequality (7) is valid for x > 1. Refinements and generalizations of (7) were given in =-=[7, 14, 16]-=- and related references therein. For more information, please refer to [13] and related references therein. In [10], by employing the monotonicity of θ(x), it was recovered simply that the double ineq... |

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Citation Context ...= t z−1 e −t dt (4) 0 for ℜz > 0. The logarithmic derivative of Γ(z), denoted by ψ(z) = Γ′ (z) Γ(z) , is called the psi or digamma function, and ψ (k) for k ∈ N are called the polygamma functions. In =-=[3]-=-, the function θ(x) = x[ln x − ψ(x)] (5) was proved to be decreasing and convex on (0, ∞), with two limits were presented complicatedly. lim θ(x) = 1 and x→0 + lim 1 θ(x) = x→∞ 2 (6) 2000 Mathematics ... |

18 |
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Citation Context ...that a function f on [0, ∞) is completely monotonic if and only if there exists a bounded and non-decreasing function α(t) such that ∫ ∞ f(x) = e −xt dα(t) (2) 0 converges for x ∈ [0, ∞). Recall also =-=[4, 5, 8, 15, 17]-=- 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. It was ... |

17 |
A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality
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- 2007
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Citation Context ...τ 120x 3 1 1 < lnx − ψ(x) < (7) 2x x for x > 0 is concluded. This extends a result in [12], which says that the inequality (7) is valid for x > 1. Refinements and generalizations of (7) were given in =-=[7, 14, 16]-=- and related references therein. For more information, please refer to [13] and related references therein. In [10], by employing the monotonicity of θ(x), it was recovered simply that the double ineq... |

16 | Logarithmically completely monotonic functions relating to the gamma function
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Citation Context ...≤ (24) (k − 1)! k! + xk xk+1 (25) Proof. In [14], the function ψ(x) −ln x+ α x was proved to be completely monotonic on (0, ∞) if and only if α ≥ 1 and so was its negative if and only if α ≤ 1 2 . In =-=[6]-=-, the function exΓ(x) xx−α was proved to be logarithmically completely monotonic on (0, ∞) if and only if α ≥ 1 and so was its reciprocal if and only if α ≤ 1 2 . From these, considering (1) and (3), ... |

16 |
A stochastic approach to the gamma function
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Citation Context ...τ 120x 3 1 1 < lnx − ψ(x) < (7) 2x x for x > 0 is concluded. This extends a result in [12], which says that the inequality (7) is valid for x > 1. Refinements and generalizations of (7) were given in =-=[7, 14, 16]-=- and related references therein. For more information, please refer to [13] and related references therein. In [10], by employing the monotonicity of θ(x), it was recovered simply that the double ineq... |

15 | Integral representation of some functions related to the gamma function
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Citation Context ...that a function f on [0, ∞) is completely monotonic if and only if there exists a bounded and non-decreasing function α(t) such that ∫ ∞ f(x) = e −xt dα(t) (2) 0 converges for x ∈ [0, ∞). Recall also =-=[4, 5, 8, 15, 17]-=- 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. It was ... |

15 |
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Citation Context ...that a function f on [0, ∞) is completely monotonic if and only if there exists a bounded and non-decreasing function α(t) such that ∫ ∞ f(x) = e −xt dα(t) (2) 0 converges for x ∈ [0, ∞). Recall also =-=[4, 5, 8, 15, 17]-=- 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. It was ... |

12 |
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Citation Context ... in [12], which says that the inequality (7) is valid for x > 1. Refinements and generalizations of (7) were given in [7, 14, 16] and related references therein. For more information, please refer to =-=[13]-=- and related references therein. In [10], by employing the monotonicity of θ(x), it was recovered simply that the double inequality xx−γ xx−1/2 < Γ(x) < ex−1 ex−1 (8) holds for x > 1, the constants γ ... |

4 |
Monotonicity and logarithmic convexity of three functions involving exponential function
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(Show Context)
Citation Context ...]∣ ∣t=∞ t=0 − ∫ ∞ 0 = 1 ∫ ∞ 1 + h 2x x ′ (t)e −xt dt. Multiplying on all sides of (29) by x and rearranging gives x[ln x − ψ(x)] = − 1 2 + ∫ ∞ 0 0 h ′ (t)e −xt } dt (28) (29) h ′ (t)e −xt dt. (30) In =-=[9, 11, 20]-=- and related references therein, the function h(t) was shown to be decreasing on (−∞, ∞), concave on (−∞, 0) and convex on (0, ∞). This means that the function θ1(x) is completely monotonic on (0, ∞) ... |

4 |
Monotonicity and logarithmic concavity of two functions involving exponential function
- Liu, Li, et al.
(Show Context)
Citation Context ...]∣ ∣t=∞ t=0 − ∫ ∞ 0 = 1 ∫ ∞ 1 + h 2x x ′ (t)e −xt dt. Multiplying on all sides of (29) by x and rearranging gives x[ln x − ψ(x)] = − 1 2 + ∫ ∞ 0 0 h ′ (t)e −xt } dt (28) (29) h ′ (t)e −xt dt. (30) In =-=[9, 11, 20]-=- and related references therein, the function h(t) was shown to be decreasing on (−∞, ∞), concave on (−∞, 0) and convex on (0, ∞). This means that the function θ1(x) is completely monotonic on (0, ∞) ... |

4 |
A concise proof for properties of three functions involving the exponential function, Appl. Math. E-Notes 9 (2009), in press
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(Show Context)
Citation Context ...]∣ ∣t=∞ t=0 − ∫ ∞ 0 = 1 ∫ ∞ 1 + h 2x x ′ (t)e −xt dt. Multiplying on all sides of (29) by x and rearranging gives x[ln x − ψ(x)] = − 1 2 + ∫ ∞ 0 0 h ′ (t)e −xt } dt (28) (29) h ′ (t)e −xt dt. (30) In =-=[9, 11, 20]-=- and related references therein, the function h(t) was shown to be decreasing on (−∞, ∞), concave on (−∞, 0) and convex on (0, ∞). This means that the function θ1(x) is completely monotonic on (0, ∞) ... |

2 | Refinements and sharpenings of some double inequalities for bounding the gamma function
- Guo, Zhang, et al.
(Show Context)
Citation Context ... (7) is valid for x > 1. Refinements and generalizations of (7) were given in [7, 14, 16] and related references therein. For more information, please refer to [13] and related references therein. In =-=[10]-=-, by employing the monotonicity of θ(x), it was recovered simply that the double inequality xx−γ xx−1/2 < Γ(x) < ex−1 ex−1 (8) holds for x > 1, the constants γ and 1 2 are the best possible, the left-... |

1 |
Some inequalities involving (r
- Minc, Sathre
(Show Context)
Citation Context ... 1) + 1 and θ(x) = 1 2 for x > 0 and τ ∈ (0, 1). From (6) and the decreasing monotonicity of θ(x), the inequality τ 120x 3 1 1 < lnx − ψ(x) < (7) 2x x for x > 0 is concluded. This extends a result in =-=[12]-=-, which says that the inequality (7) is valid for x > 1. Refinements and generalizations of (7) were given in [7, 14, 16] and related references therein. For more information, please refer to [13] and... |