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16
On some inequalities for the gamma and psi functions
 MATH. COMP
, 1997
"... We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, starshaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) = ..."
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Cited by 78 (3 self)
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We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, starshaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) =
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 27 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
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 21 (14 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
, 2009
"... 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 func ..."
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Cited by 18 (15 self)
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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.
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 ..."
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Cited by 18 (18 self)
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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 ..."
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Cited by 14 (2 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.
Summations for Basic Hypergeometric Series Involving a QAnalogue of the Digamma Function
, 1996
"... Using a simple method, numerous summation formulas for hypergeometric and basic hypergeometric series are derived. Among these summation formulas are nonterminating extensions and qextensions of identities recorded by Lavoie, Luke, Watson, and Srivastava. At the result side of the basic hypergeomet ..."
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Cited by 8 (1 self)
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Using a simple method, numerous summation formulas for hypergeometric and basic hypergeometric series are derived. Among these summation formulas are nonterminating extensions and qextensions of identities recorded by Lavoie, Luke, Watson, and Srivastava. At the result side of the basic hypergeometric summations there appears a qanalogue of the digamma function. Some of its properties are also studied. 1.
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. ..."
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Cited by 5 (1 self)
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Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and qgamma 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 i ..."
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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
Inequalities for the Gamma function via convexity
 in P. Cerone and S. Dragomir (eds), Advances in Inequalities for Special Functions, Nova Science Publishers
"... Abstract. We review techniques based on convexity, logarithmic convexity and Schurconvexity, for producing inequalities and asymptotic expansions for ratios of Gamma functions. As an illustration, results for the Gautschi’s and Gurland’s ratio are presented, as well as asymptotic expansions for th ..."
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Abstract. We review techniques based on convexity, logarithmic convexity and Schurconvexity, for producing inequalities and asymptotic expansions for ratios of Gamma functions. As an illustration, results for the Gautschi’s and Gurland’s ratio are presented, as well as asymptotic expansions for the Gamma function, along the lines of W. Krull’s work. We argue that convexitybased techniques are advantageous over other methods, because they enable a comparision of inequalities, provide two transformations for their sharpening, and also yield two sided asymptotic expansions. The Gamma function... is simple enough for juniors in college to meet, but deep enough to have called forth contributions from the finest mathematicians. Philip Davis [7] 1