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14
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
NECESSARY AND SUFFICIENT CONDITIONS FOR A FUNCTION INVOLVING DIVIDED DIFFERENCES OF THE DI AND TRIGAMMA FUNCTIONS TO BE COMPLETELY MONOTONIC
, 903
"... Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two g ..."
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Cited by 7 (7 self)
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Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions. 1.
Some properties of the psi and polygamma functions, Available online at http://arxiv.org/abs/0903.1003
"... Abstract. In this paper, some monotonicity and concavity results of several functions involving the psi and polygamma functions are proved, and then some known inequalities are extended and generalized. 1. ..."
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Cited by 4 (4 self)
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Abstract. In this paper, some monotonicity and concavity results of several functions involving the psi and polygamma functions are proved, and then some known inequalities are extended and generalized. 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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Cited by 4 (4 self)
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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.
Complete monotonicity of some functions involving polygamma functions, submitted
"... Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) ..."
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Cited by 3 (3 self)
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Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) (x) are polygamma functions. 1.
A COMPLETE SOLUTION TO AN OPEN PROBLEM RELATING TO AN INEQUALITY FOR RATIOS OF GAMMA FUNCTIONS
, 902
"... Abstract. In this paper, we prove that for x + y> 0 and y + 1> 0 the inequality [Γ(x + y + 1)/Γ(y + 1)] 1/x s x + y [Γ(x + y + 2)/Γ(y + 1)] 1/(x+1) x + y + 1 is valid if x> 1 and reversed if x < 1, where Γ(x) is the Euler gamma function. This completely extends the result in [Y. Yu, An inequality fo ..."
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Cited by 2 (2 self)
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Abstract. In this paper, we prove that for x + y> 0 and y + 1> 0 the inequality [Γ(x + y + 1)/Γ(y + 1)] 1/x s x + y [Γ(x + y + 2)/Γ(y + 1)] 1/(x+1) x + y + 1 is valid if x> 1 and reversed if x < 1, where Γ(x) is the Euler gamma function. This completely extends the result in [Y. Yu, An inequality for ratios of gamma
Supplements to a class of logarithmically completely monotonic functions associated with the gamma function
 Appl. Math. Comput
"... Abstract. In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on (0, ∞). As applications of the necessary and sufficient condition, some inequalities for bounding the ps ..."
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Cited by 1 (1 self)
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Abstract. In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on (0, ∞). As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived. This is a continuator of the paper [12]. 1.
LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS INVOLVING p − GAMMA FUNCTIONS
"... Abstract. In this paper we prove that the function fα,β,p(x) = Γp(x + β) px (x + xx+β−α () 2, and ..."
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Abstract. In this paper we prove that the function fα,β,p(x) = Γp(x + β) px (x + xx+β−α () 2, and
(k − 1)! h (k−1)!
, 903
"... Abstract. The main aim of this paper is to prove that the double inequality ..."
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Abstract. The main aim of this paper is to prove that the double inequality
TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION
, 902
"... Abstract. In the present paper, we give two new proofs for the necessary and sufficient condition α ≤ 1 such that the function x α [lnx − ψ(x)] is completely monotonic on (0, ∞). 1. ..."
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Abstract. In the present paper, we give two new proofs for the necessary and sufficient condition α ≤ 1 such that the function x α [lnx − ψ(x)] is completely monotonic on (0, ∞). 1.