Results 1 
7 of
7
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 ..."
Abstract

Cited by 21 (14 self)
 Add to MetaCart
(Show Context)
Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x
A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a powerexponential function
, 2009
"... 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 functio ..."
Abstract

Cited by 18 (18 self)
 Add to MetaCart
(Show Context)
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 14 (2 self)
 Add to MetaCart
(Show Context)
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.
An interesting double inequality for Euler’s gamma function
 J. Inequal. Pure Appl. Math
"... ABSTRACT. In this short paper we derive new and interesting upper and lower bounds for the Euler’s gamma function in terms of the digamma function ψ(x) = Γ ′ (x)/Γ(x). ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
ABSTRACT. In this short paper we derive new and interesting upper and lower bounds for the Euler’s gamma function in terms of the digamma function ψ(x) = Γ ′ (x)/Γ(x).
Some monotonicity properties of gamma and qgamma functions, Available onlie at http://arxiv.org/abs/0709.1126v2
"... Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and qgamma functions. 1. ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and qgamma functions. 1.
Asymptotic approximations to the HardyLittlewood function
, 2012
"... The function Q(x):= n≥1(1/n) sin(x/n) was introduced by Hardy and Littlewood [10] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos [1 ..."
Abstract
 Add to MetaCart
The function Q(x):= n≥1(1/n) sin(x/n) was introduced by Hardy and Littlewood [10] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos [1] of a conjecture by Clark and Ismail [2]. More precisely, Alzer et. al. have shown that the Clark and Ismail conjecture is true if and only if Q(x) ≥ −pi/2 for all x> 0. It is known that Q(x) is unbounded in the domain x ∈ (0,∞) from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point x for which Q(x) < −pi/2. This turns out to be a surprisingly hard problem, which leads to an interesting and nontrivial question of how to approximate Q(x) for very large values of x. In this paper we continue the work started by Gautschi in [7] and develop several approximations to Q(x) for large values of x. We use these approximations to find an explicit value of x for which Q(x) < −pi/2.