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A Logarithmically Completely monotonic Function Involving the Gamma Functions 1
"... We show that the function x → [Γ(x+1)]1/x x[Γ(x+2)] 1/(x+1) is logarithmically completely monotonic on (0, ∞). This answers a question by A.Vernescu. ..."
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We show that the function x → [Γ(x+1)]1/x x[Γ(x+2)] 1/(x+1) is logarithmically completely monotonic on (0, ∞). This answers a question by A.Vernescu.
Some logarithmically completely monotonic functions involving gamma function
, 2005
"... Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x ..."
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Cited by 10 (6 self)
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Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x
On powers of Stieltjes moment sequences
, 2005
"... For a Bernstein function f the sequence sn = f(1)·...·f(n) is a Stieltjes moment sequence with the property that all powers s c n, c> 0 are again Stieltjes moment sequences. We prove that s c n is Stieltjes determinate for c ≤ 2, but it can be indeterminate for c> 2 as is shown by the moment sequenc ..."
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Cited by 8 (4 self)
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For a Bernstein function f the sequence sn = f(1)·...·f(n) is a Stieltjes moment sequence with the property that all powers s c n, c> 0 are again Stieltjes moment sequences. We prove that s c n is Stieltjes determinate for c ≤ 2, but it can be indeterminate for c> 2 as is shown by the moment sequence (n!) c, corresponding to the Bernstein function f(s) = s. Nevertheless there always exists a unique product convolution semigroup (ρc)c>0 such that ρc has moments s c n. We apply the indeterminacy of (n!) c for c> 2 to prove that the distribution of the product of p independent identically distributed normal random variables is indeterminate if and only if p ≥ 3.
Turán type inequalities for hypergeometric functions
 Proc. Amer. Math. Soc
"... Dedicated to the memory of Professor Alexandru Lupa¸s Abstract. In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some ..."
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Cited by 4 (1 self)
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Dedicated to the memory of Professor Alexandru Lupa¸s Abstract. In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems. 1.
A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a powerexponential function, submitted
 CLASS OF COMPLETELY MONOTONIC FUNCTIONS AND APPLICATIONS 11
"... 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 ..."
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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.
Porcu The Dagum family of isotropic correlation functions available at arXiv:0705.0456v1 [math.ST
"... A function ρ:[0, ∞) → (0,1] is a completely monotonic function if and only if ρ(‖x ‖ 2) is positive definite on R d for all d and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they ..."
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Cited by 3 (0 self)
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A function ρ:[0, ∞) → (0,1] is a completely monotonic function if and only if ρ(‖x ‖ 2) is positive definite on R d for all d and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function ( β x ρ(β,γ)(x) = 1 − 1 + xβ)γ, x ≥ 0, β,γ> 0, called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any ddimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.
StieltjesPickBernsteinSchoenberg and their connection to complete monotonicity
, 2007
"... This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: Rk, [0, ∞ [ k, N0–the first with the inverse involution and the two others wit ..."
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Cited by 3 (1 self)
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This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: Rk, [0, ∞ [ k, N0–the first with the inverse involution and the two others with the identical involution. Schoenberg’s theorem explains the possibility of constructing rotation invariant positive definite and conditionally negative definite functions on euclidean spaces via completely monotonic functions and Bernstein functions. It is therefore important to be able to decide complete monotonicity of a given function. We combine complete monotonicity with complex analysis via the relation to Stieltjes functions and Pick functions and we give a survey of the many interesting relations between these classes of functions and completely monotonic functions, logarithmically completely monotonic functions and Bernstein functions. In Section 6 it is proved that log x − Ψ(x) and Ψ ′ (x) are logarithmically completely monotonic (where Ψ(x) = Γ ′ (x)/Γ(x)), and these results are new as far as we know. We end with a list of completely monotonic functions related to the Gamma function.
HOW MANY LAPLACE TRANSFORMS OF PROBABILITY MEASURES ARE THERE?
, 2010
"... A bracketing metric entropy bound for the class of Laplace transforms of probability measures on [0, ∞) is obtained through its connection with the small deviation probability of a smooth Gaussian process. Our results for the particular smooth Gaussian process seem to be of independent interest. ..."
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Cited by 1 (1 self)
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A bracketing metric entropy bound for the class of Laplace transforms of probability measures on [0, ∞) is obtained through its connection with the small deviation probability of a smooth Gaussian process. Our results for the particular smooth Gaussian process seem to be of independent interest.
Stieltjes and other integral representations for functions of
"... ( v2.0 released June 2011) We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci & Sokal. In addition, we consider the relationship o ..."
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( v2.0 released June 2011) We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci & Sokal. In addition, we consider the relationship of functions of W to the class of completely monotonic functions and show that W is a complete Bernstein function.