## (k − 1)! h (k−1)! (903)

### BibTeX

@MISC{Qi903(k−,

author = {Feng Qi and Bai-ni Guo and Ψ(k I/k},

title = {(k − 1)! h (k−1)!},

year = {903}

}

### OpenURL

### Abstract

Abstract. The main aim of this paper is to prove that the double inequality

### Citations

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Some completely monotonic functions involving polygamma functions and an application
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Citation Context ... 2n+1 1 ∑ B2j ln(2π) + 2 2 2j(2j − 1)x j=1 2j−1 (15) are completely monotonic on (0, ∞). In [26, Theorem 1], the convexity of the functions Fn(x) and Gn(x) were presented alternatively. Stimulated by =-=[42]-=-, the complete monotonicity of Fn(x) and Gn(x) were simply verified in [25, Theorem 2] again. 1.5. The second kind of inequalities for the psi and polygamma functions. In [19, Theorem 1], the function... |

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Citation Context ...ly monotonic functions are also completely monotonic, but not conversely. This result was formally published when revising [41]. Hereafter, this conclusion and its proofs were dug in [12, 15, 16] and =-=[58]-=- (the preprint of [46]) once and again. Furthermore, in the paper [12], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely monotonic... |

19 | Sharp inequalities for the digamma and polygamma functions - Alzer |

19 |
Complete monotonicities of functions involving the gamma and digamma functions
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Citation Context ...perties of completely monotonic functions can be found in [27, Chapter XIII], [59, Chapter IV] and closely-related references therein. 1.2. Logarithmically completely monotonic functions. Recall also =-=[6, 47]-=- that a function f is said to be logarithmically completely monotonic on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm lnf satisfies 0 ≤ (−1) k [lnf(x)] (k) < ∞ (3) for ... |

18 | Some classes of logarithmically completely monotonic functions involving gamma function, submitted - Qi, Guo |

17 | Integral representation of some functions related to the gamma function
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Citation Context ...hmically completely monotonic functions are also completely monotonic, but not conversely. This result was formally published when revising [41]. Hereafter, this conclusion and its proofs were dug in =-=[12, 15, 16]-=- and [58] (the preprint of [46]) once and again. Furthermore, in the paper [12], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely ... |

17 | The best bounds in Gautschi’s inequality - Elezović, Giordano, et al. |

17 | Note on a class of completely monotonic functions involving the polygamma functions, Tamsui Oxf
- Qi, Guo, et al.
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Citation Context ...ved to be logarithmically completely monotonic on (0, ∞) if and only if α ≥ 1, so is its reciprocal if and only if α ≤ 1 2 . From these, the following double inequalities were derived and employed in =-=[17, 36, 40, 44, 45, 48, 49, 50, 52, 56, 57]-=-: For x ∈ (0, ∞) and k ∈ N, we have lnx − 1 1 < ψ(x) < lnx − x 2x (5) and (k − 1)! k! + xk 2xk+1 < ∣ ∣ (k) ψ (x) (k − 1)! k! < + . xk xk+1 (6) In [3, Theorem 9], it was proved that if k ≥ 1 and n ≥ 0 ... |

16 | Logarithmically completely monotonic functions relating to the gamma function - Chen, Qi |

16 |
A complete monotonicity property of the gamma function
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Citation Context ...l or not? In [47, Theorem 4], it was proved that all logarithmically completely monotonic functions are also completely monotonic, but not conversely. This result was formally published when revising =-=[41]-=-. Hereafter, this conclusion and its proofs were dug in [12, 15, 16] and [58] (the preprint of [46]) once and again. Furthermore, in the paper [12], the logarithmically completely monotonic functions ... |

14 |
Completely monotonic functions involving the gamma and q-gamma functions
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Citation Context ...hmically completely monotonic functions are also completely monotonic, but not conversely. This result was formally published when revising [41]. Hereafter, this conclusion and its proofs were dug in =-=[12, 15, 16]-=- and [58] (the preprint of [46]) once and again. Furthermore, in the paper [12], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely ... |

14 | Some monotonicity properties and characterizations of the gamma function - Muldoon - 1978 |

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12 |
Some properties of a class of logarithmically completely monotonic functions
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Citation Context ...perties of completely monotonic functions can be found in [27, Chapter XIII], [59, Chapter IV] and closely-related references therein. 1.2. Logarithmically completely monotonic functions. Recall also =-=[6, 47]-=- that a function f is said to be logarithmically completely monotonic on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm lnf satisfies 0 ≤ (−1) k [lnf(x)] (k) < ∞ (3) for ... |

12 |
On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlich- keitstheorie und Verw
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Citation Context ...) once and again. Furthermore, in the paper [12], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely monotonic functions studied in =-=[22]-=- and all Stieltjes transforms were proved to be logarithmically completely monotonic on (0, ∞). For more information, please refer to [12]. 1.3. The gamma and polygamma functions. It is well-known tha... |

12 |
Bounds for the ratio of two gamma functions
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Citation Context ...e enough to 0, but not when x > 0 is large enough. For more information on further investigation of functions similar to (16), please refer to the research papers [18, 20, 21], the expository article =-=[32]-=- and related references therein. 1.6. A sharp inequality for the psi function and related results. In [8, Lemma 1.7] and [51, Theorem 1], it was proved that the double inequality ( ln x + 1 ) − 2 1 x ... |

12 |
The best bounds in Kershaw’s inequality and two completely monotonic functions
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Citation Context ... and n ∈ N. This inequality can be restated more meaningfully as √ ∣∣ψ n+1 (n+1)(x) ∣ < n! n √∣∣ψ (n)(x) ∣ . (26) (n − 1)! In [5, Lemma 4.6], the inequality (22) was generalized to the q-analogue. In =-=[31, 37]-=-, the preprints of [43, 53], the divided difference ⎧[ ] 2 ⎪⎨ ψ(x + t) − ψ(x + s) + ∆s,t(x) = t − s ⎪⎩ ψ′ (x + t) − ψ ′(x + s) , s ̸= t t − s [ψ ′ (x + s)] 2 + ψ ′′ (x + s), s = t for |t − s| < 1 and ... |

11 | Some new inequalities for gamma and polygamma functions - Batir - 2005 |

10 | A class of logarithmically completely monotonic functions
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Citation Context ... inequality in (6) when x > 0 is close enough to 0, but not when x > 0 is large enough. For more information on further investigation of functions similar to (16), please refer to the research papers =-=[18, 20, 21]-=-, the expository article [32] and related references therein. 1.6. A sharp inequality for the psi function and related results. In [8, Lemma 1.7] and [51, Theorem 1], it was proved that the double ine... |

10 |
On Wallis’ formula
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Citation Context ...) is completely monotonic if and only if λ < 1, (b) so is the function −∆s,t;λ(x) if and only if λ > 1, (c) and ∆s,t;1(x) ≡ 0.6 F. QI AND B.-N. GUO These results generalize the claim in the proof of =-=[24]-=-. For detailed information, see related texts remarked in the expository article [33]. In [9, Remark 2.3], it was pointed out that the inequality ψ ′′ (x) + [ ψ ′ ( x + 1 2 )] 2 < 0 (31) for x > 0 is ... |

9 | On some properties of digamma and polygamma function - Batir |

7 | Logarithmic convexity and inequalities for the gamma function - Merkle - 1996 |

7 | Necessary and sufficient conditions for a function involving a ratio of gamma functions to be logarithmically completely monotonic, Available online at http: //arxiv.org/abs/0904.1101
- Qi, Guo
(Show Context)
Citation Context ...ved to be logarithmically completely monotonic on (0, ∞) if and only if α ≥ 1, so is its reciprocal if and only if α ≤ 1 2 . From these, the following double inequalities were derived and employed in =-=[17, 36, 40, 44, 45, 48, 49, 50, 52, 56, 57]-=-: For x ∈ (0, ∞) and k ∈ N, we have lnx − 1 1 < ψ(x) < lnx − x 2x (5) and (k − 1)! k! + xk 2xk+1 < ∣ ∣ (k) ψ (x) (k − 1)! k! < + . xk xk+1 (6) In [3, Theorem 9], it was proved that if k ≥ 1 and n ≥ 0 ... |

6 | Grinshpan, Inequalities for the gamma and q−gamma functions - Alzer, Z |

5 | An interesting double inequality for Euler’s gamma function - Batir |

5 | Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic, Integral Transforms Spec - GUO, QI, et al. |

5 | Bounds for the ratio of two gamma functions—From Wendel’s limit to ElezovićGiordano-Pečarić’s theorem, Available online at http://arxiv.org/abs/0902.2514
- Qi
(Show Context)
Citation Context ...and only if λ > 1, (c) and ∆s,t;1(x) ≡ 0.6 F. QI AND B.-N. GUO These results generalize the claim in the proof of [24]. For detailed information, see related texts remarked in the expository article =-=[33]-=-. In [9, Remark 2.3], it was pointed out that the inequality ψ ′′ (x) + [ ψ ′ ( x + 1 2 )] 2 < 0 (31) for x > 0 is a direct consequence of [9, Theorem 2.2]: For x > 0, 1 ≤ k ≤ n − 1 and n ∈ N, we have... |

4 |
Inequalities for the Gamma function relating to asymptotic expansions
- Allasia, Giordano, et al.
- 2002
(Show Context)
Citation Context ...i+k (8) [ k−1 with the usual convention that an empty sum is nil and Bi for i ≥ 0 are Bernoulli numbers defined by t et − 1 = ∞∑ t Bi i x = 1 − i! 2 + ∞∑ x B2j 2j , |x| < 2π. (9) (2j)! x > 1 2 i=0 In =-=[2]-=-, among other things, the following double inequalities were procured: For , we have 2N+1 ( ) ∑ 1 B2k 2 k=1 2k ( x − 1 ( ) < ln x − 2k 2 1 ) 2N∑ ( ) 1 B2k 2 − ψ(x) < 2 k=1 2k ( x − 1 ) (10) 2k 2 and j... |

4 | Remarks on some completely monotonic functions - Koumandos |

4 | A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Available online at http://arxiv.org/abs/0903.5038
- Qi, Guo
(Show Context)
Citation Context ... are also completely monotonic, but not conversely. This result was formally published when revising [41]. Hereafter, this conclusion and its proofs were dug in [12, 15, 16] and [58] (the preprint of =-=[46]-=-) once and again. Furthermore, in the paper [12], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely monotonic functions studied in ... |

4 | Sharp inequalities for the psi function and harmonic numbers, Available online at http://arxiv.org/abs/0902.2524
- Qi, Guo
(Show Context)
Citation Context ...tions with (5) as (17) does. More strongly, the function Q(x) = e ψ(x+1) − x (20) was proved in [51, Theorem 2] to be strictly decreasing and convex on (−1, ∞) with . The basic tools of the proofs in =-=[51]-=- include limx→∞ Q(x) = 1 2 and ψ ′ (x)e ψ(x) < 1, x > 0 (21) [ψ ′ (x)] 2 + ψ ′′ (x) > 0, x > 0. (22) Among other things, the monotonicity and convexity of the function (20) were also derived in [14, C... |

4 | Some properties of the psi and polygamma functions, submitted
- Qi, Guo
(Show Context)
Citation Context ...ved to be logarithmically completely monotonic on (0, ∞) if and only if α ≥ 1, so is its reciprocal if and only if α ≤ 1 2 . From these, the following double inequalities were derived and employed in =-=[17, 36, 40, 44, 45, 48, 49, 50, 52, 56, 57]-=-: For x ∈ (0, ∞) and k ∈ N, we have lnx − 1 1 < ψ(x) < lnx − x 2x (5) and (k − 1)! k! + xk 2xk+1 < ∣ ∣ (k) ψ (x) (k − 1)! k! < + . xk xk+1 (6) In [3, Theorem 9], it was proved that if k ≥ 1 and n ≥ 0 ... |

3 | Inequalities for the Gamma function - Batir - 2008 |

3 | Some monotonicity properties of gamma and q-gamma functions, Available onlie at http://arxiv.org/abs/0709.1126
- Gao
(Show Context)
Citation Context ...hmically completely monotonic functions are also completely monotonic, but not conversely. This result was formally published when revising [41]. Hereafter, this conclusion and its proofs were dug in =-=[12, 15, 16]-=- and [58] (the preprint of [46]) once and again. Furthermore, in the paper [12], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely ... |

3 |
A new proof of complete monotonicity of a function involving psi function
- Qi, Guo
(Show Context)
Citation Context |

3 | Complete monotonicity of some functions involving polygamma functions, Available online at http://arxiv.org/abs/0905.2732
- Qi, Guo, et al.
(Show Context)
Citation Context |

2 |
Complete monotonicity results of a function involving the divided difference of the psi functions and consequences, submitted
- Qi, Guo
(Show Context)
Citation Context ...mpletely monotonic if α ≥ sup x∈(0,∞) x φ−1 ([2(x + 1) 2 − 1]e2x , (34) ) where φ −1 denotes the inverse function of φ(x) = xcoth x on (0, ∞). In passing, it is noted that the results demonstrated in =-=[29, 30, 35, 40]-=- have very close relations with the above mentioned conclusions. 1.7. Main results of this paper. The main aim of this paper is to sharpen the double inequality (18) and to generalize the sharp inequa... |

2 |
Complete monotonicity results of divided difference of psi functions and new bounds for ratio of two gamma functions, submitted
- Qi, Cerone, et al.
(Show Context)
Citation Context |

1 | Supplements to a class of logarithmically completely monotonic functions associated with the gamma function
- Guo, Qi, et al.
(Show Context)
Citation Context ... inequality in (6) when x > 0 is close enough to 0, but not when x > 0 is large enough. For more information on further investigation of functions similar to (16), please refer to the research papers =-=[18, 20, 21]-=-, the expository article [32] and related references therein. 1.6. A sharp inequality for the psi function and related results. In [8, Lemma 1.7] and [51, Theorem 1], it was proved that the double ine... |

1 | Complete monotonicity of a polygamma function plus the square of another polygamma function, submitted
- Qi, Guo
(Show Context)
Citation Context |