## NECESSARY AND SUFFICIENT CONDITIONS FOR A FUNCTION INVOLVING DIVIDED DIFFERENCES OF THE DI- AND TRI-GAMMA FUNCTIONS TO BE COMPLETELY MONOTONIC (903)

Citations: | 7 - 7 self |

### BibTeX

@MISC{Qi903necessaryand,

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

title = {NECESSARY AND SUFFICIENT CONDITIONS FOR A FUNCTION INVOLVING DIVIDED DIFFERENCES OF THE DI- AND TRI-GAMMA FUNCTIONS TO BE COMPLETELY MONOTONIC},

year = {903}

}

### OpenURL

### Abstract

Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions. 1.

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Citation Context ...ory, combinatorics, physics, engineering, and other mathematical sciences. In particular, the functions ψ(x) and ψ ′ (x) for x > 0 are also called the digamma and trigamma functions respectively, see =-=[1]-=- and [13, p. 71]. By using the double inequalities 1 1 1 1 + + − x 2x2 6x3 30x5 < ψ′ (x) < 1 x see [17, p. 860, Theorem 4], and + 1 2x 1 + , (5) 2 6x3 − 1 1 1 − − x2 x3 2x4 < ψ′′ (x) < − 1 1 − , (6) x... |

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Citation Context ...unctions. The first author was partially supported by the China Scholarship Council. This paper was typeset using AMS-LATEX. 12 F. QI AND B.-N. GUO basic properties of LCM functions, please refer to =-=[12, 18, 37]-=- and related references therein. It is well-known that the classical Euler’s gamma function ∫ ∞ Γ(x) = t x−1 e −t dt (4) 0 for x > 0, the psi function ψ(x) = Γ′ (x) Γ(x) and the polygamma functions ψ(... |

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Citation Context ...ly monotonic functions are also completely monotonic, but not conversely. This result was formally published while revising [30]. Hereafter, this conclusion and its proofs were dug in [6, 11, 12] and =-=[49]-=- (the preprint of [33]) once and again. Furthermore, in the paper [6], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely monotonic ... |

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Citation Context ...ed and non-decreasing function α(t) such that ∫ ∞ f(x) = e −xt dα(t) (2) 0 converges for x ∈ [0, ∞). This expresses that a CM function f on [0, ∞) is a Laplace transform of the measure α. Recall also =-=[6, 31]-=- that a function f is said to be logarithmically completely monotonic (LCM) on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm lnf satisfies (−1) k [lnf(x)] (k) ≥ 0 (3) fo... |

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

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Citation Context ...unction ⎧[ ] 2 ⎪⎨ ψ(x + t) − ψ(x + s) + ∆s,t(x) = t − s ⎪⎩ ψ′ (x + t) − ψ ′(x + s) , s ̸= t t − s (19) [ψ ′ (x + s)] 2 + ψ ′′ (x + s), s = t for |t − s| < 1 and −∆s,t(x) for |t − s| > 1 are proved in =-=[23, 26]-=- to be CM on (−α, ∞). Using the complete monotonicity of the function (19), the inequality (11) and [5, Theorem 4.3] mentioned on page 3 were generalized in [29, Theorem 5] to a4 F. QI AND B.-N. GUO ... |

17 | A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality - QI, GUO - 2007 |

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Citation Context ...tonic on (0, ∞), i.e., ψ(x) − lnx + α x (−1) i[ ψ(x) − lnx + α x ] (i) ≥ 0 (16) for i ≥ 0, if and only if α ≥ 1, so is its negative, i.e., the inequality (16) is reversed, if and only if α ≤ 1 2 . In =-=[8]-=- and [20, Theorem 2.1], the function exΓ(x) xx−α was proved to be logarithmically completely monotonic on (0, ∞), i.e., (−1) k [ ln exΓ(x) xx−α ](k) ≥ 0 (17) for k ∈ N, if and only if α ≥ 1, so is its... |

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Citation Context ...unctions. The first author was partially supported by the China Scholarship Council. This paper was typeset using AMS-LATEX. 12 F. QI AND B.-N. GUO basic properties of LCM functions, please refer to =-=[12, 18, 37]-=- and related references therein. It is well-known that the classical Euler’s gamma function ∫ ∞ Γ(x) = t x−1 e −t dt (4) 0 for x > 0, the psi function ψ(x) = Γ′ (x) Γ(x) and the polygamma functions ψ(... |

15 | Chángyòng Bùděngshì (Applied Inequalities), 3rd ed., Shāndōng Kēxué Jìshù Chūbǎn Shè (Shandong Science and - KUANG - 2004 |

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Citation Context ... or not? In [35, Theorem 4], it was proved that all logarithmically completely monotonic functions are also completely monotonic, but not conversely. This result was formally published while revising =-=[30]-=-. Hereafter, this conclusion and its proofs were dug in [6, 11, 12] and [49] (the preprint of [33]) once and again. Furthermore, in the paper [6], the logarithmically completely monotonic functions on... |

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Citation Context ...unctions. The first author was partially supported by the China Scholarship Council. This paper was typeset using AMS-LATEX. 12 F. QI AND B.-N. GUO basic properties of LCM functions, please refer to =-=[12, 18, 37]-=- and related references therein. It is well-known that the classical Euler’s gamma function ∫ ∞ Γ(x) = t x−1 e −t dt (4) 0 for x > 0, the psi function ψ(x) = Γ′ (x) Γ(x) and the polygamma functions ψ(... |

12 |
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Citation Context ...unction ⎧[ ] 2 ⎪⎨ ψ(x + t) − ψ(x + s) + ∆s,t(x) = t − s ⎪⎩ ψ′ (x + t) − ψ ′(x + s) , s ̸= t t − s (19) [ψ ′ (x + s)] 2 + ψ ′′ (x + s), s = t for |t − s| < 1 and −∆s,t(x) for |t − s| > 1 are proved in =-=[23, 26]-=- to be CM on (−α, ∞). Using the complete monotonicity of the function (19), the inequality (11) and [5, Theorem 4.3] mentioned on page 3 were generalized in [29, Theorem 5] to a4 F. QI AND B.-N. GUO ... |

11 |
Some properties of a class of logarithmically completely monotonic functions
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- 1988
(Show Context)
Citation Context ...ed and non-decreasing function α(t) such that ∫ ∞ f(x) = e −xt dα(t) (2) 0 converges for x ∈ [0, ∞). This expresses that a CM function f on [0, ∞) is a Laplace transform of the measure α. Recall also =-=[6, 31]-=- that a function f is said to be logarithmically completely monotonic (LCM) on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm lnf satisfies (−1) k [lnf(x)] (k) ≥ 0 (3) fo... |

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

11 |
Bounds for the Ratio of Two Gamma Functions
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Citation Context ...y procured in [16, Theorem 1], several alternative proofs were supplied in [14, 23, 26, 30, 39, 40]. The investigation of the function (18) has a long history, see [20, 21, 41] or the survey articles =-=[24, 25]-=- and related references therein. Acknowledgements. This manuscript was completed during the first author’s visit to the RGMIA, Victoria University, Australia, between March 2008 and February 2009. The... |

10 |
The best bounds in Gautschi-Kershaw inequalities
- QI, GUO, et al.
(Show Context)
Citation Context ...2) or the increasing property of φ(x). Remark 5. After the monotonic and convex properties of the function (18) were perfectly procured in [16, Theorem 1], several alternative proofs were supplied in =-=[14, 23, 26, 30, 39, 40]-=-. The investigation of the function (18) has a long history, see [20, 21, 41] or the survey articles [24, 25] and related references therein. Acknowledgements. This manuscript was completed during the... |

10 |
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Citation Context ...]) once and again. Furthermore, in the paper [6], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely monotonic functions studied in =-=[16]-=- and all Stieltjes transforms were proved to be logarithmically completely monotonic on (0, ∞). For more information, please refer to [6]. It is well-known that the classical Euler gamma function Γ(x)... |

10 | A class of logarithmically completely monotonic functions
- Guo, Srivastava
(Show Context)
Citation Context ...asing with respect to x ≥ 1 for fixed y ≥ 0. Consequently, for positive real numbers x ≥ 1 and y ≥ 0, we have x + y + 1 x + y + 2 ≤ [Γ(x + y + 1)/Γ(y + 1)]1/x . (4) [Γ(x + y + 2)/Γ(y + 1)] 1/(x+1) In =-=[32]-=-, the above decreasing monotonicity was extended and generalized as follows: The function (3) is logarithmically completely monotonic with respect to x ∈ (0, ∞) if y ≥ 0, so is its reciprocal if −1 < ... |

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

9 |
New upper bounds in the second Kershaw’s double inequality and its generalizations
- Qi, Guo
(Show Context)
Citation Context ... Lemma 3 is thus proved. □ Remark 6. It is noted that the left-hand side inequality (27) for i = 0 and p = −1 contains (24). For more information about the inequalities (24) and (27), please refer to =-=[23, 24, 44, 46, 47]-=- and related references therein. 4. Proof of Theorem 1 For x = 0, taking the logarithm of hα,y(x) gives ln Γ(x + y + 1) − ln Γ(y + 1) lnhα,y(x) = − α ln(x + y + 1). x Direct differentiation yields k∑... |

8 |
On Wallis’ formula
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- 1956
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Citation Context ...e function Γ(x+s) Γ(x+t) on (−t, ∞) is increasingly convex for s−t > 1 and increasingly concave for 0 < s−t < 1. For detailed information, please refer to [25, Remark 4.2 and Remark 4.5] on the paper =-=[20]-=-. Remark 2. From the proofs of Theorem 1 and Theorem 2, the following conclusions may be summarized: For real numbers s, t, α = min{s, t} and λ, the function ⎧ ψ(x + t) − ψ(x + s) 1 + λ(2x + s + t) ⎪⎨... |

8 |
Inequalities for the volume of unit ball in R n
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Citation Context ... setting different values of x, y and t in (7), more similar inequalities as above may be derived immediately. For more information on inequalities for the volume of the unit ball in R n , please see =-=[2, 3, 36]-=- and related references therein. Remark 3. It is noted that an alternative upper bound in (4) and (7) for t = 1 has been established in [34] and related references therein. Remark 4. It is noted that ... |

8 |
Inequalities and monotonicity for the ratio of gamma functions
- GUO, QI
(Show Context)
Citation Context ...xp[ψ(y + 1)], x = 0. (y + 1) α ] 1/x , x ∈ (−y − 1, ∞) \ {0}; It is clear that the ranges of x, y and α in the function hα,y(x) extend the corresponding ones in the functions (3) and (5) discussed in =-=[13, 32, 41, 53]-=-. The aim of this paper is to present necessary and sufficient conditions such that the function (6) or its reciprocal are logarithmically completely monotonic. Our main results may be stated as follo... |

8 | extensions and generalizations of the second Kershaw’s double inequality - Qi, Refinements |

7 | A Dictionary of - BULLEN - 1998 |

7 |
A note on gamma function
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- 1959
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Citation Context ... 1 and a = x + 1 2 yields [ ] 2 ( Γ(x + 1) < x + Γ(x + 1/2) 1 )√ x + 1/2 2 x + 1 for x > − 1 2 , which is a refinement of the inequality [ ] 2 Γ(x + 1) − x < Γ(x + 1/2) 1 2 for x > −1 2 , obtained in =-=[41]-=-. (50) (51)12 F. QI AND B.-N. GUO Remark 4. In [10, Lemma 1.2] and [11, Lemma 1.2], it was discovered that if a ≤ − ln 2 and b ≥ 0, then a − ln ( e 1/x − 1 ) < ψ(x) < b − ln ( e 1/x − 1 ) (52) holds ... |

7 |
Note on the gamma function
- Wendel
- 1948
(Show Context)
Citation Context ...y virtue of (36). Consequently, the necessities for the function Hs,t;λ(x) to be LCM on (−α, ∞) are verified. The left proofs are similar and so omitted. Theorem 2 is proved. □ Proof of Theorem 3. In =-=[42]-=-, the following asymptotic relation was obtained: Γ(x + s) lim x→∞ xs = 1 (44) Γ(x) for real s and x holds. This implies that x + s Hs,t;λ(x) = [(x + s)(x + t)] λ/2 [ Γ(x + t) (x + s) t−s ] 1/(t−s) 1/... |

7 | A new lower bound in the second Kershaw’s double inequality - Qi - 1016 |

7 |
Some inequalities involving geometric mean of natural numbers and ratio of gamma functions
- QI, GUO
(Show Context)
Citation Context ...a function, inequality. The second author was partially supported by the China Scholarship Council. This paper was typeset using AMS-LATEX. 12 F. QI AND B.-N. GUO In [13, Theorem 2] and its preprint =-=[41]-=-, the following monotonicity was established: The function [Γ(x + y + 1)/Γ(y + 1)] 1/x (3) x + y + 1 is decreasing with respect to x ≥ 1 for fixed y ≥ 0. Consequently, for positive real numbers x ≥ 1 ... |

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

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

5 | Monotonicity and convexity for the gamma function
- Chen
(Show Context)
Citation Context ...2) or the increasing property of φ(x). Remark 5. After the monotonic and convex properties of the function (18) were perfectly procured in [16, Theorem 1], several alternative proofs were supplied in =-=[14, 23, 26, 30, 39, 40]-=-. The investigation of the function (18) has a long history, see [20, 21, 41] or the survey articles [24, 25] and related references therein. Acknowledgements. This manuscript was completed during the... |

5 |
Functional equations for Wallis and Gamma
- Lazarević, Lupa¸s
- 1974
(Show Context)
Citation Context ...s of the function (18) were perfectly procured in [16, Theorem 1], several alternative proofs were supplied in [14, 23, 26, 30, 39, 40]. The investigation of the function (18) has a long history, see =-=[20, 21, 41]-=- or the survey articles [24, 25] and related references therein. Acknowledgements. This manuscript was completed during the first author’s visit to the RGMIA, Victoria University, Australia, between M... |

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 ...y procured in [16, Theorem 1], several alternative proofs were supplied in [14, 23, 26, 30, 39, 40]. The investigation of the function (18) has a long history, see [20, 21, 41] or the survey articles =-=[24, 25]-=- and related references therein. Acknowledgements. This manuscript was completed during the first author’s visit to the RGMIA, Victoria University, Australia, between March 2008 and February 2009. The... |

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

4 |
Sharp inequalities for the harmonic numbers
- Alzer
(Show Context)
Citation Context ...s discovered that if a ≤ − ln 2 and b ≥ 0, then a − ln ( e 1/x − 1 ) < ψ(x) < b − ln ( e 1/x − 1 ) (52) holds for x > 0. In [9, Theorem 2.8], the inequality (52) was sharpened as a ≤ −γ and b ≥ 0. In =-=[4]-=-, the function φ(x) defined by (13) was proved to be strictly increasing on (0, ∞) and lim φ(x) = 0. (53) x→∞ In [35], among other things, the function φ(x) was proved to be both strictly increasing a... |

4 | Sharp inequalities for the psi function and harmonic numbers, Available online at http://arxiv.org/abs/0902.2524 - Qi, Guo |

4 | Some properties of the psi and polygamma functions, submitted
- Qi, Guo
(Show Context)
Citation Context ...> 0. In [9, Theorem 2.8], the inequality (52) was sharpened as a ≤ −γ and b ≥ 0. In [4], the function φ(x) defined by (13) was proved to be strictly increasing on (0, ∞) and lim φ(x) = 0. (53) x→∞ In =-=[35]-=-, among other things, the function φ(x) was proved to be both strictly increasing and concave on (0, ∞), with limx→0 + φ(x) = −γ and the limit (53). It is not difficult to see that all these results e... |

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 while revising [30]. Hereafter, this conclusion and its proofs were dug in [6, 11, 12] and [49] (the preprint of =-=[33]-=-) once and again. Furthermore, in the paper [6], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely monotonic functions studied in [... |

4 |
Monotonicity of sequences involving convex function and sequence
- QI
(Show Context)
Citation Context ... to be logarithmically concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if 0 ≤ α ≤ 1 4 . For more information on the history, background, motivation and generalizations of this topic, please refer to =-=[1, 5, 14, 25, 26, 37, 38, 41, 45, 52]-=- and related references therein. For given y ∈ (−1, ∞) and α ∈ (−∞, ∞), let ⎧ ⎪⎨ 1 hα,y(x) = (x + y + 1) ⎪⎩ α [ Γ(x + y + 1) Γ(y + 1) 1 exp[ψ(y + 1)], x = 0. (y + 1) α ] 1/x , x ∈ (−y − 1, ∞) \ {0}; I... |

3 | An alternative proof of Elezović-Giordano-Pečarić’s theorem, Available online at http://arxiv.org/abs/0903.1174
- Qi, Guo
(Show Context)
Citation Context ...2) or the increasing property of φ(x). Remark 5. After the monotonic and convex properties of the function (18) were perfectly procured in [16, Theorem 1], several alternative proofs were supplied in =-=[14, 23, 26, 30, 39, 40]-=-. The investigation of the function (18) has a long history, see [20, 21, 41] or the survey articles [24, 25] and related references therein. Acknowledgements. This manuscript was completed during the... |