## A completely monotone function related to the Gamma function (1999)

Venue: | J. Comp. Appl. Math |

Citations: | 16 - 6 self |

### BibTeX

@ARTICLE{Berg99acompletely,

author = {Christian Berg and Henrik L. Pedersen},

title = {A completely monotone function related to the Gamma function},

journal = {J. Comp. Appl. Math},

year = {1999},

volume = {133},

pages = {219--230}

}

### OpenURL

### Abstract

Abstract We show that the reciprocal of the function f (z) = log \Gamma (z + 1) z log z; z 2 C n] \Gamma 1; 0];

### Citations

241 |
The classical moment problem and some related questions in analysis
- Akhiezer
- 1965
(Show Context)
Citation Context ...+ 1 n2 ' \Gammasx n ' = \Gammaslog \Gamma (x + 1) + 1 2 1X n=1 log ` 1 + 1 n2 ' : As noted before, the function j\Gamma (x+ iy + 1)j is decreasing for positive y. Its values are all ^ 1 since 1 + x 2 =-=[1; 2]-=-. Therefore 0 ^ log 1 j\Gamma (z + 1)j ^ log 1 j\Gamma (x + i + 1)j ^ log 1 \Gamma (x + 1) + 1 2 1X n=1 log ` 1 + 1 n2 ' ^ 1 5 + 1 2 1X n=1 1 n2 = 1 5 + ss2 12 : Here we have also used that log \Gamma... |

115 |
Potential Theory on Locally Compact Abelian Groups
- Berg, Forst
- 1975
(Show Context)
Citation Context ...t. Non-negative functions on the half-line ]0; 1[ with a completely monotone derivative appears in the literature under the names of completely monotone mappings, cf. [7] and Bernstein functions, cf. =-=[6]-=-. Theorem 1.1 can be rephrased that log \Gamma (x + 1)=x log x is a Bernstein function. There is an important relation between the class S and the class B of Bernstein functions. We state this relatio... |

82 |
Harmonic Analysis and the Theory of Probability
- Bochner
- 1955
(Show Context)
Citation Context ...ous functions ' of compact support. Non-negative functions on the half-line ]0; 1[ with a completely monotone derivative appears in the literature under the names of completely monotone mappings, cf. =-=[7]-=- and Bernstein functions, cf. [6]. Theorem 1.1 can be rephrased that log \Gamma (x + 1)=x log x is a Bernstein function. There is an important relation between the class S and the class B of Bernstein... |

55 | The Logarithmic Integral - KOOSIS - 1988 |

28 |
Handbuch der Theorie der Gammafunktion
- Nielsen
- 1906
(Show Context)
Citation Context ...Px(y) j fly + 1X n=1 ` arctan ` y n + x ' \Gammasy n ' = \Gammasarg \Gamma (z + 1): (10) It is known that the convex function log \Gamma (x + 1) has its minimum on [0; 1[ at a point x0 ss 0:461, c.f. =-=[10]-=-. We have log \Gamma (x0 + 1) ss \Gamma 0:121 ? \Gamma 1=5. Since log \Gamma (z + 1) is the complex conjugate of log \Gamma (z + 1) for z 2 A, it is enough to prove that log \Gamma (z + 1) 6= 0; for z... |

26 |
A monotonicity property of the gamma function
- ANDERSON, QIU
- 1997
(Show Context)
Citation Context ... x ? 0 has attracted the attention of several authors. A similar function (where \Gamma (x + 1) is replaced by \Gamma (1 + x=2)) occurred in the paper of Anderson, Vamanamurthy and Vuorinen ([4]). In =-=[3]-=-, Anderson and Qiu showed that f increases on the interval [1; 1[ and they conjectured that f is concave on the interval [1; 1[. The concavity of f on [1; 1[ was established by Elbert and Laforgia ([8... |

22 |
On some properties of the gamma function
- ELBERT, LAFORGIA
(Show Context)
Citation Context ...3], Anderson and Qiu showed that f increases on the interval [1; 1[ and they conjectured that f is concave on the interval [1; 1[. The concavity of f on [1; 1[ was established by Elbert and Laforgia (=-=[8]-=-). At the 1sconference in Patras in September 1999, the following conjecture about f was made: for every n * 1, the inequality (\Gamma 1) n\Gamma 1 f (n) (x) * 0 holds for x 2 [1; 1[. The purpose of t... |

12 |
Quelques remarques sur le cône de Stieltjes. - In: S¶eminaire de Th¶eorie du Potentiel
- Berg
- 1980
(Show Context)
Citation Context ...1; 0]: Proposition 1.2 Let ~S denote the set of holomorphic functions G : A ! C satisfying 2s(i) =G(z) ^ 0 for =z ? 0, (ii) G(x) * 0 for x ? 0. Then fGj]0; 1[ : G 2 ~Sg = S. A proof is written out in =-=[5]-=-, which also contains a list of stability properties of the cone S. The constant a in (1) is clearly given as a = lim x!1 g(x): The measure _ can be found from the holomorphic extension of (1) to A gi... |

7 |
Special functions of quasiconformal theory
- ANDERSON, VAMANAMURTHY, et al.
- 1989
(Show Context)
Citation Context ...x log x ; x ? 0 has attracted the attention of several authors. A similar function (where \Gamma (x + 1) is replaced by \Gamma (1 + x=2)) occurred in the paper of Anderson, Vamanamurthy and Vuorinen (=-=[4]-=-). In [3], Anderson and Qiu showed that f increases on the interval [1; 1[ and they conjectured that f is concave on the interval [1; 1[. The concavity of f on [1; 1[ was established by Elbert and Laf... |

1 | Geometric properties of the Gamma function
- Ahern, Rudin
- 1996
(Show Context)
Citation Context ...2 A, it is enough to prove that log \Gamma (z + 1) 6= 0; for z = x + iy; y ? 0: (11) This will be done in five steps: (i) The function log \Gamma (z + 1) is univalent in the half-plane f!z ? x0g, cf. =-=[1]-=-, and since it vanishes at z = 1, it does not vanish elsewhere in f!z ? x0g. (ii) In the strip 0 ! !z ^ x0 we know that j\Gamma (z + 1)j ^ \Gamma (x + 1) ! 1, and hence ! log \Gamma (z + 1) = log j\Ga... |

1 | le d'eveloppement de log \Gamma (a - Stieltjes, Sur - 1993 |

1 | le développement de log Γ(a - Stieltjes, Sur - 1993 |