## SOME PROPERTIES OF THE GAMMA AND PSI FUNCTIONS, WITH APPLICATIONS

Citations: | 6 - 0 self |

### BibTeX

@MISC{Qiu_someproperties,

author = {S. -l. Qiu and M. Vuorinen},

title = {SOME PROPERTIES OF THE GAMMA AND PSI FUNCTIONS, WITH APPLICATIONS},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume Ωn of the unit ball B n ⊂ R n,thesurface area ωn−1 of the unit sphere S n−1, and some related constants. 1.

### Citations

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Citation Context ...as � ∞ Γ(x) = (1.1) 0 t x−1 e −t dt, B(x, y) = Γ(x)Γ(y) Γ(x + y) , ψ(x) =Γ′ (x) Γ(x) , respectively. For the extensions to complex variables and for the basic properties of these functions, see [AS], =-=[AAR]-=-, [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], [G], [Ke], [K2], [L], [MSC] ... |

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99 |
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Citation Context ... (1.1) 0 t x−1 e −t dt, B(x, y) = Γ(x)Γ(y) Γ(x + y) , ψ(x) =Γ′ (x) Γ(x) , respectively. For the extensions to complex variables and for the basic properties of these functions, see [AS], [AAR], [Mi], =-=[T]-=-, and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], [G], [Ke], [K2], [L], [MSC] and bibliog... |

52 |
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Citation Context .../(n − 1) and cn =(2Jn) 1−n (1.5) ωn−2, for n ∈ N with n ≥ 2. These constants have applications in some fields of mathematics such as geometry of Grassmannian subspaces of Rn [KR], optimization theory =-=[B]-=-, geometric function theory [AVV1, pp. 234–246], [V], [Vu], as well as geometry of spaces of constant curvature [BH]. These are some examples of the fields where the gamma function is frequently used.... |

42 | On some inequalities for the gamma and psi functions
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Citation Context ...ties of these functions, see [AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], =-=[A2]-=-, [A5], [G], [Ke], [K2], [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studie... |

29 |
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Citation Context ...ctions, see [AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], [G], =-=[Ke]-=-, [K2], [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in [AQ... |

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Citation Context ...properties of these functions, see [AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see =-=[A1]-=-, [A2], [A5], [G], [Ke], [K2], [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been ... |

25 |
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Citation Context ...Ke], [K2], [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in =-=[AQ]-=-, [A1]–[A4], [BP], [EL]. Let Bn and Sn−1 be the unit ball and unit sphere in Rn , respectively, Ωn be the volume of Bn ,andωn−1denote the surface area of Sn−1 . Set Ω0 =1. ItiswellknownthatΩnis increa... |

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Citation Context ...rrell [TT] obtained the bounds (2.9) 1 2(n +1) <dn − γ< 1 , n ≥ 2, 2(n − 1) for dn − γ, which was improved by R. M. Young [Y] as 1 2(n +1) <dn− γ< 1 , for n ∈ N, 2n (2.10) while G. D. Anderson et al. =-=[ABRVV]-=- proved 1 − γ n <dn− γ< 1 , for n ∈ N. 2n (2.11) Recently, H. Alzer [A3, Theorem 3] established the double inequality 1 2(n + α) ≤ dn 1 − γ< , for n ∈ N, 2(n + β) (2.12) where α = {1/[2(1 − γ)]}−1=0.1... |

20 |
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Citation Context ...nd bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in [AQ], [A1]–[A4], [BP], =-=[EL]-=-. Let Bn and Sn−1 be the unit ball and unit sphere in Rn , respectively, Ωn be the volume of Bn ,andωn−1denote the surface area of Sn−1 . Set Ω0 =1. ItiswellknownthatΩnis increasing for 2 ≤ n ≤ 5 and ... |

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Citation Context ...e functions, see [AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], =-=[G]-=-, [Ke], [K2], [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently ... |

15 | A completely monotone function related to the gamma function
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Citation Context ...MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in [AQ], [A1]–[A4], =-=[BP]-=-, [EL]. Let Bn and Sn−1 be the unit ball and unit sphere in Rn , respectively, Ωn be the volume of Bn ,andωn−1denote the surface area of Sn−1 . Set Ω0 =1. ItiswellknownthatΩnis increasing for 2 ≤ n ≤ ... |

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Citation Context ... 2. These constants have applications in some fields of mathematics such as geometry of Grassmannian subspaces of Rn [KR], optimization theory [B], geometric function theory [AVV1, pp. 234–246], [V], =-=[Vu]-=-, as well as geometry of spaces of constant curvature [BH]. These are some examples of the fields where the gamma function is frequently used. The accumulated literature on the gamma function is so va... |

8 |
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Citation Context ...[L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in [AQ], [A1]–=-=[A4]-=-, [BP], [EL]. Let Bn and Sn−1 be the unit ball and unit sphere in Rn , respectively, Ωn be the volume of Bn ,andωn−1denote the surface area of Sn−1 . Set Ω0 =1. ItiswellknownthatΩnis increasing for 2 ... |

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Citation Context ...[AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], [G], [Ke], [K2], =-=[L]-=-, [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in [AQ], [A1]–[A4... |

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Citation Context ...h n ≥ 2. These constants have applications in some fields of mathematics such as geometry of Grassmannian subspaces of Rn [KR], optimization theory [B], geometric function theory [AVV1, pp. 234–246], =-=[V]-=-, [Vu], as well as geometry of spaces of constant curvature [BH]. These are some examples of the fields where the gamma function is frequently used. The accumulated literature on the gamma function is... |

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Citation Context ...thematics such as geometry of Grassmannian subspaces of Rn [KR], optimization theory [B], geometric function theory [AVV1, pp. 234–246], [V], [Vu], as well as geometry of spaces of constant curvature =-=[BH]-=-. These are some examples of the fields where the gamma function is frequently used. The accumulated literature on the gamma function is so vast that it is difficult for someone working chiefly in the... |

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Citation Context ..., −α(x)+(1− s)θ(x)/s} ω(s, x) =min{sη(x) − β(x), −α(x)}. The above theorem yields some inequalities of Γ(x). Some similar or related results have been proved recently in [AQ], [BP], [EL], [A5], [K2], =-=[Me1]-=-–[Me3]. 1.15. Remarks. (1) For n ∈ N, letG(n, k) be the so-called Grassmannian, andlet τn denote the invariant measure on G(n, 1), that is, on the set of all straight lines through the origin (cf. [KR... |

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Citation Context ...sing for n ∈ N since g(n) = � nπ/2f1(n/2) by the expression of g(n) (see [KR, p. 214]).s726 S.-L. QIU AND M. VUORINEN (2) The double inequality (1.14) and its Corollary 3.2 are related to a result in =-=[LL]-=-, which gives the estimates � (1 − s) log(x + s/2) <Ds(x) < (1 − s)log x +(Γ(s)) 1/(s−1)� for s ∈ (0, 1) and x>0, where Ds(x) =logΓ(x +1)−log Γ(x + s). For some other related results, see [A3, p. 365]... |

6 | Convexity, schur-convexity and bounds for the gamma function involving the digamma function - Merkle - 1998 |

6 |
Conditions for convexity of a derivative and some applications to the Gamma function
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Citation Context ...)+(1− s)θ(x)/s} ω(s, x) =min{sη(x) − β(x), −α(x)}. The above theorem yields some inequalities of Γ(x). Some similar or related results have been proved recently in [AQ], [BP], [EL], [A5], [K2], [Me1]–=-=[Me3]-=-. 1.15. Remarks. (1) For n ∈ N, letG(n, k) be the so-called Grassmannian, andlet τn denote the invariant measure on G(n, 1), that is, on the set of all straight lines through the origin (cf. [KR, p. 2... |

5 |
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Citation Context ...f these functions, see [AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], =-=[A5]-=-, [G], [Ke], [K2], [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied rece... |

5 | Sur la fonction Gamma, Publ - SÁNDOR - 1989 |

4 | Inequalities for the gamma and polygamma - ALZER - 1998 |

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Citation Context ...orem 3.1]. (2) It is well known that [AS, 6.1.3] (2.8) and by [AS, 6.3.2], γ = lim n→∞ dn, dn = n� k=1 1 − log n, k dn − γ = ψ(n +1)− log n.s730 S.-L. QIU AND M. VUORINEN S. R. Tims and J. A. Tyrrell =-=[TT]-=- obtained the bounds (2.9) 1 2(n +1) <dn − γ< 1 , n ≥ 2, 2(n − 1) for dn − γ, which was improved by R. M. Young [Y] as 1 2(n +1) <dn− γ< 1 , for n ∈ N, 2n (2.10) while G. D. Anderson et al. [ABRVV] pr... |

3 |
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Citation Context ... ≤ 1 β − , 2n n2 (2.14) with equality iff n = 1, where the constants α = 1/12 = 0.08333 ··· and β = γ − 1/2 =0.07721 ··· are best possible. Some further results on the approximation of γ are given in =-=[K1]-=-. 3. Proofs of the main theorems In this section we prove the theorems stated in Section 1. 3.1. Proof of Theorem 1.12. (1) Logarithmic differentiation gives f ′ 1 (x)/f1(x) =h1(x), where h1 is as in ... |

2 |
Topics in special functions, Papers on Analysis: A volume dedicated to Olli Martio on the occasion of his 60th birthday
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Citation Context ...ction, our main research area being geometric function theory; see [V], [Vu], [AVV1], [AQ]. Some of the latest developments on special functions that have been studied in this context are reviewed in =-=[AVV2]-=-. The statement of some of our main results, which we think have independent interest as such, now follows. Theorem A. For n =1, 2, 3, ... � � π Ωn π 2 ≤ < 2 2n + π − 2 Ωn−1 2n +1 . Theorem B. For x ≥... |

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Citation Context ..., see [AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], [G], [Ke], =-=[K2]-=-, [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in [AQ], [A1... |

1 |
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Citation Context ...n−1 cn � 1/(n−1) , bn = Jn/(n − 1) and cn =(2Jn) 1−n (1.5) ωn−2, for n ∈ N with n ≥ 2. These constants have applications in some fields of mathematics such as geometry of Grassmannian subspaces of Rn =-=[KR]-=-, optimization theory [B], geometric function theory [AVV1, pp. 234–246], [V], [Vu], as well as geometry of spaces of constant curvature [BH]. These are some examples of the fields where the gamma fun... |

1 |
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Citation Context ...t x−1 e −t dt, B(x, y) = Γ(x)Γ(y) Γ(x + y) , ψ(x) =Γ′ (x) Γ(x) , respectively. For the extensions to complex variables and for the basic properties of these functions, see [AS], [AAR], [Mi], [T], and =-=[WW]-=-. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], [G], [Ke], [K2], [L], [MSC] and bibliographies th... |

1 |
President’s Office, Hangzhou Institute of Electronics Engineering (HIEE), Hangzhou 310037, Peoples Republic of China E-mail address: sl qiu@hziee.edu.cn
- Gaz
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Citation Context ...dn − γ = ψ(n +1)− log n.s730 S.-L. QIU AND M. VUORINEN S. R. Tims and J. A. Tyrrell [TT] obtained the bounds (2.9) 1 2(n +1) <dn − γ< 1 , n ≥ 2, 2(n − 1) for dn − γ, which was improved by R. M. Young =-=[Y]-=- as 1 2(n +1) <dn− γ< 1 , for n ∈ N, 2n (2.10) while G. D. Anderson et al. [ABRVV] proved 1 − γ n <dn− γ< 1 , for n ∈ N. 2n (2.11) Recently, H. Alzer [A3, Theorem 3] established the double inequality ... |