## A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, submitted

Venue: | CLASS OF COMPLETELY MONOTONIC FUNCTIONS AND APPLICATIONS 11 |

Citations: | 5 - 5 self |

### BibTeX

@INPROCEEDINGS{Qi_aproperty,

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

title = {A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, submitted},

booktitle = {CLASS OF COMPLETELY MONOTONIC FUNCTIONS AND APPLICATIONS 11},

year = {}

}

### OpenURL

### Abstract

Abstract. In the article, a notion “logarithmically absolutely monotonic function” is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function α x+β 1+ are proved, where α and β are given real parameters, a new proof x for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed.

### Citations

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Citation Context ...given real parameters, a new proof x for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed. 1. Introduction Recall =-=[32, 33, 35, 52, 54]-=- that a function f is said to be completely monotonic on an interval I if f has derivative of all orders on I such that (−1) k f (k) (x) ≥ 0 (1) for x ∈ I and k ≥ 0. For our own convenience, the set o... |

58 | The curious history of Faà di Bruno’s formula, AMS Monthly 109
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Citation Context ...if and only if 2β ≥ α. In [7] and [38, 41], two different proofs for the inclusion CL[I] ⊂ C[I] were given. Now we would like to present a new proof for this inclusion by using Faá di Bruno’s formula =-=[16, 24, 53]-=-. Theorem 4 ([7, 38, 41]). A logarithmically completely monotonic function on an interval I is also completely monotonic on I, but not conversely. Equivalently, CL[I] ⊂ C[I] and C[I] \ CL[I] ̸= ∅. Now... |

46 | On some inequalities for the gamma and psi functions
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(Show Context)
Citation Context ... monotonic, maybe it is sufficient and much simpler to prove their logarithmically complete monotonicity or to show that they are Stieltjes transforms, if possible. These techniques have been used in =-=[1, 2, 3, 5, 9, 10, 11, 12, 13, 14, 15, 17, 22, 28, 29, 30, 31, 32, 33, 36, 37, 40, 43, 44, 45, 48, 49, 50, 52, 56]-=- and many other articles. It can be imagined that, if there is no the inclusion relationships between the sets of completely monotonic functions, logarithmically completely monotonic functions and Sti... |

33 |
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Citation Context ...atives of all orders and f (k−1) (t) ≥ 0 (2) for t ∈ I and k ∈ N, where N denotes the set of all positive integers. The set of the absolutely monotonic functions on I is denoted by A[I]. Recall again =-=[6, 32, 33, 38, 41, 43, 44]-=- that a positive function f is said to be logarithmically completely monotonic on an interval I if its logarithm lnf satisfies (−1) k [lnf(x)] (k) ≥ 0 (3) for k ∈ N on I. Similar to above, the set of ... |

28 |
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Citation Context ...much simpler or easier to prove the stronger statement that they are logarithmically absolutely monotonic. Let ( Fα,β(x) = 1 + α ) x+β (6) x for α ̸= 0 and either x > max{0, −α} or x < min{0, −α}. In =-=[18, 19, 20, 23, 25, 34, 39, 42, 46, 47, 55]-=- (See also the related contents in [26, 27]), the sufficient and necessary conditions such that the function Fα,β(x), its simplified forms, its variants and their corresponding sequences are monotonic... |

27 |
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Citation Context ... monotonic, maybe it is sufficient and much simpler to prove their logarithmically complete monotonicity or to show that they are Stieltjes transforms, if possible. These techniques have been used in =-=[1, 2, 3, 5, 9, 10, 11, 12, 13, 14, 15, 17, 22, 28, 29, 30, 31, 32, 33, 36, 37, 40, 43, 44, 45, 48, 49, 50, 52, 56]-=- and many other articles. It can be imagined that, if there is no the inclusion relationships between the sets of completely monotonic functions, logarithmically completely monotonic functions and Sti... |

26 |
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Citation Context ... monotonic, maybe it is sufficient and much simpler to prove their logarithmically complete monotonicity or to show that they are Stieltjes transforms, if possible. These techniques have been used in =-=[1, 2, 3, 5, 9, 10, 11, 12, 13, 14, 15, 17, 22, 28, 29, 30, 31, 32, 33, 36, 37, 40, 43, 44, 45, 48, 49, 50, 52, 56]-=- and many other articles. It can be imagined that, if there is no the inclusion relationships between the sets of completely monotonic functions, logarithmically completely monotonic functions and Sti... |

22 |
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Citation Context ...atives of all orders and f (k−1) (t) ≥ 0 (2) for t ∈ I and k ∈ N, where N denotes the set of all positive integers. The set of the absolutely monotonic functions on I is denoted by A[I]. Recall again =-=[6, 32, 33, 38, 41, 43, 44]-=- that a positive function f is said to be logarithmically completely monotonic on an interval I if its logarithm lnf satisfies (−1) k [lnf(x)] (k) ≥ 0 (3) for k ∈ N on I. Similar to above, the set of ... |

20 |
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Citation Context ...ND B.-N. GUO converges for 0 < x < ∞, and that f(x) ∈ A[(0, ∞)] if and only if there exists a bounded and nondecreasing function σ(t) such that f(x) = ∫ ∞ 0 e xt dσ(t) (5) converges for 0 ≤ x < ∞. In =-=[7, 38, 41, 43, 44, 52]-=- and many other references, the inclusions CL[I] ⊂ C[I] and S ⊂ CL[(0, ∞)] were revealed implicitly or explicitly, where S denotes the class of Stieltjes transforms. The class CL[(0, ∞)] is characteri... |

18 |
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Citation Context |

17 |
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17 |
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Citation Context ..., a further deep investigation on the logarithmically completely monotonic functions was explicitly carried out in [7] and a citation of the logarithmically completely monotonic functions appeared in =-=[17]-=-. It is said in [7] that “In various papers complete monotonicity for special functions has been established by proving the stronger statement that the function is a Stieltjes transform”. It is also s... |

17 |
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Citation Context |

17 |
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(Show Context)
Citation Context ...given real parameters, a new proof x for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed. 1. Introduction Recall =-=[32, 33, 35, 52, 54]-=- that a function f is said to be completely monotonic on an interval I if f has derivative of all orders on I such that (−1) k f (k) (x) ≥ 0 (1) for x ∈ I and k ≥ 0. For our own convenience, the set o... |

16 | Some classes of completely monotonic functions
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(Show Context)
Citation Context ...∞)] for α > 0 and β ∈ R if and only if 2β ≥ α > 0. From CL[I] ⊂ C[I] it is deduced that the function ( 1 + α ) x+β − e x α ∈ C[(0, ∞)] (7) if and only if 0 < α ≤ 2β, which is a conclusion obtained in =-=[4]-=-. Now it is natural to pose a problem: How about the logarithmically complete or absolute monotonicity of the function Fα,β(x) for all real numbers α ̸= 0 and β in the interval (−∞, min{0, −α}) or (ma... |

16 | A completely monotone function related to the Gamma function
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Citation Context ... said in [7] that “In various papers complete monotonicity for special functions has been established by proving the stronger statement that the function is a Stieltjes transform”. It is also said in =-=[8]-=- that “In concrete cases it is often easier to establish that a function is a Stieltjes transform than to verify complete monotonicity”. Because the logarithmically completely monotonic functions must... |

16 | Logarithmically completely monotonic functions relating to the gamma function
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(Show Context)
Citation Context |

16 |
Chángyòng Bùděngsh̀ı (Applied Inequalities), 3rd ed., Shandong Science and
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Citation Context ...cally absolutely monotonic. Let ( Fα,β(x) = 1 + α ) x+β (6) x for α ̸= 0 and either x > max{0, −α} or x < min{0, −α}. In [18, 19, 20, 23, 25, 34, 39, 42, 46, 47, 55] (See also the related contents in =-=[26, 27]-=-), the sufficient and necessary conditions such that the function Fα,β(x), its simplified forms, its variants and their corresponding sequences are monotonic are obtained. In [48, Theorem 1.2], [49, T... |

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Citation Context ...atives of all orders and f (k−1) (t) ≥ 0 (2) for t ∈ I and k ∈ N, where N denotes the set of all positive integers. The set of the absolutely monotonic functions on I is denoted by A[I]. Recall again =-=[6, 32, 33, 38, 41, 43, 44]-=- that a positive function f is said to be logarithmically completely monotonic on an interval I if its logarithm lnf satisfies (−1) k [lnf(x)] (k) ≥ 0 (3) for k ∈ N on I. Similar to above, the set of ... |

12 | On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlich- keitstheorie und Verw - Horn - 1976 |

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Citation Context ...if and only if 2β ≥ α. In [7] and [38, 41], two different proofs for the inclusion CL[I] ⊂ C[I] were given. Now we would like to present a new proof for this inclusion by using Faá di Bruno’s formula =-=[16, 24, 53]-=-. Theorem 4 ([7, 38, 41]). A logarithmically completely monotonic function on an interval I is also completely monotonic on I, but not conversely. Equivalently, CL[I] ⊂ C[I] and C[I] \ CL[I] ̸= ∅. Now... |

11 | Logarithmically completely monotonic functions involving gamma and polygamma functions
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8 |
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Citation Context ...given real parameters, a new proof x for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed. 1. Introduction Recall =-=[32, 33, 35, 52, 54]-=- that a function f is said to be completely monotonic on an interval I if f has derivative of all orders on I such that (−1) k f (k) (x) ≥ 0 (1) for x ∈ I and k ≥ 0. For our own convenience, the set o... |

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Citation Context ...cally absolutely monotonic. Let ( Fα,β(x) = 1 + α ) x+β (6) x for α ̸= 0 and either x > max{0, −α} or x < min{0, −α}. In [18, 19, 20, 23, 25, 34, 39, 42, 46, 47, 55] (See also the related contents in =-=[26, 27]-=-), the sufficient and necessary conditions such that the function Fα,β(x), its simplified forms, its variants and their corresponding sequences are monotonic are obtained. In [48, Theorem 1.2], [49, T... |

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Citation Context ...much simpler or easier to prove the stronger statement that they are logarithmically absolutely monotonic. Let ( Fα,β(x) = 1 + α ) x+β (6) x for α ̸= 0 and either x > max{0, −α} or x < min{0, −α}. In =-=[18, 19, 20, 23, 25, 34, 39, 42, 46, 47, 55]-=- (See also the related contents in [26, 27]), the sufficient and necessary conditions such that the function Fα,β(x), its simplified forms, its variants and their corresponding sequences are monotonic... |

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Citation Context ...f has derivative of all orders on I such that (−1) k f (k) (x) ≥ 0 (1) for x ∈ I and k ≥ 0. For our own convenience, the set of the completely monotonic functions on I is denoted by C[I]. Recall also =-=[32, 33, 35, 51, 52, 54]-=- that a function f is said to be absolutely monotonic on an interval I if it has derivatives of all orders and f (k−1) (t) ≥ 0 (2) for t ∈ I and k ∈ N, where N denotes the set of all positive integers... |

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1 |
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Citation Context ...if and only if 2β ≥ α. In [7] and [38, 41], two different proofs for the inclusion CL[I] ⊂ C[I] were given. Now we would like to present a new proof for this inclusion by using Faá di Bruno’s formula =-=[16, 24, 53]-=-. Theorem 4 ([7, 38, 41]). A logarithmically completely monotonic function on an interval I is also completely monotonic on I, but not conversely. Equivalently, CL[I] ⊂ C[I] and C[I] \ CL[I] ̸= ∅. Now... |

1 |
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