Results 1  10
of
11
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. ..."
Abstract

Cited by 16 (12 self)
 Add to MetaCart
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.
A RESTARTED LANCZOS APPROXIMATION TO FUNCTIONS OF A SYMMETRIC MATRIX
"... In this paper, we investigate a method for restarting the Lanczos method for approximating the matrixvector product f(A)b, where A ∈ R n×n is a symmetric matrix. For analytic f we derive a novel restart function that identifies the error in the Lanczos approximation. The restart procedure is then ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
In this paper, we investigate a method for restarting the Lanczos method for approximating the matrixvector product f(A)b, where A ∈ R n×n is a symmetric matrix. For analytic f we derive a novel restart function that identifies the error in the Lanczos approximation. The restart procedure is then generated by a restart formula using a sequence of these restart functions. We present an error bound for the proposed restart scheme. We also present an error bound for the restarted Lanczos approximation of f(A)b for symmetric positive definite A when f is in a particular class of completely monotone functions. We illustrate for some important matrix function applications the usefulness of these bounds for terminating the restart process once the desired accuracy in the matrix function approximation has been achieved.
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 ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
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.
Pick functions related to the gamma function
 Conference on Special Functions
, 2000
"... ABSTRACT. We show that the function f(z) = log Γ(z +1) zLog z holomorphic in the complex plane cut along the negative real axis, is a Pick function and we find its integral representation. We also show that various other related functions are Pick functions. 1. Introduction. The ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
ABSTRACT. We show that the function f(z) = log Γ(z +1) zLog z holomorphic in the complex plane cut along the negative real axis, is a Pick function and we find its integral representation. We also show that various other related functions are Pick functions. 1. Introduction. The
SOME PROPERTIES OF THE GAMMA AND PSI FUNCTIONS, WITH APPLICATIONS
"... 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 wellknown results for the volume Ωn of the unit ball B n ⊂ R n,thesu ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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 wellknown 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.
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 ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
Monotonicity and logarithmic convexity relating to the volume of the unit ball, submitted
"... Abstract. Let Ωn stand for the volume of the unit ball in Rn for n ∈ N. In the 1/(n ln n) present paper, we prove that the sequence Ωn is logarithmically convex 1/(n ln n) Ω and that the sequence is strictly decreasing for n ≥ 2. In n Ω 1/[(n+1)ln(n+1)] n+1 addition, some monotonic and concave prope ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. Let Ωn stand for the volume of the unit ball in Rn for n ∈ N. In the 1/(n ln n) present paper, we prove that the sequence Ωn is logarithmically convex 1/(n ln n) Ω and that the sequence is strictly decreasing for n ≥ 2. In n Ω 1/[(n+1)ln(n+1)] n+1 addition, some monotonic and concave properties of several functions relating to Ωn are extended and generalized.