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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 23 (15 self)
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Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x
The extended mean values: definition, properties, monotonicities, comparison, convexities, generalizations, and applications
 No.1, Art.5. Available online at http://rgmia.vu.edu.au/v5n1.html
"... Abstract. The extended mean values E(r, s;x, y) play an important role in theory of mean values and theory of inequalities, and even in the whole mathematics, since many norms in mathematics are always means. Its study is not only interesting but important, both because most of the twovariable mea ..."
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Abstract. The extended mean values E(r, s;x, y) play an important role in theory of mean values and theory of inequalities, and even in the whole mathematics, since many norms in mathematics are always means. Its study is not only interesting but important, both because most of the twovariable mean values are special cases of E(r, s;x, y), and because it is challenging to study a function whose formulation is so indeterminate. In this expositive article, we summarize the recent main results about study of E(r, s;x, y), including definition, basic properties, monotonicities, comparison, logarithmic convexities, Schurconvexities, generalizations of concepts of mean values, applications to quantum, to theory of special functions, to establishment of Steffensen pairs, and to generalization of HermiteHadamard’s inequality. 1. Definition and expressions of the extended mean values The histories of mean values and inequalities are long [9]. The mean values are related to the Mean Value Theorems for derivative or for integral, which are the bridge between the local and global properties of functions. The arithmeticmeangeometricmean inequality is probably the most important inequality, and certainly a keystone of the theory of inequalities [2]. Inequalities of mean values are one of the main parts of theory of inequalities, they have explicit geometric meanings [14]. The theory of mean values plays an important role in the whole mathematics, since many norms in mathematics are always means. 1.1. Definition of the extended mean values. In 1975, the extended mean values E(r, s;x, y) were defined in [51] by K. B. Stolarsky as follows E(r, s;x, y) = r
Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions
 J. Inequal. Appl. 2006 (2006), Article ID
"... We denote by Γ(a) and Γ(a;z) the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to x> 0, a lower bound established by Elbert and Laforgia (2000) for the function ∫ x 0 e −tpdt = [Γ(1/p)−Γ(1/p;xp)]/p, wit ..."
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We denote by Γ(a) and Γ(a;z) the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to x> 0, a lower bound established by Elbert and Laforgia (2000) for the function ∫ x 0 e −tpdt = [Γ(1/p)−Γ(1/p;xp)]/p, with p> 1, only for 0 < x < (9(3p+1)/4(2p+1))1/p. Copyright © 2006 A. Laforgia and P. Natalini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and
Beijing, People’s Republic of China
, 2010
"... We derive saddlepoint approximations for the distribution and density functions of the halflife estimated by OLS from autoregressive timeseries models. Our results are used to prove that none of the integerorder moments of these halflife estimators exist. This provides an explanation for the ver ..."
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We derive saddlepoint approximations for the distribution and density functions of the halflife estimated by OLS from autoregressive timeseries models. Our results are used to prove that none of the integerorder moments of these halflife estimators exist. This provides an explanation for the very large estimates of persistency, and the extremely wide confidence intervals, that have been reported by various authors – for example in the empirical economics literature relating to purchasing power parity.
Some properties on the function involving the Gamma function
 Applied Mathematics. 2012
"... We studied the monotonicity and Convexity properties of the new functions involving the gamma function, and get the general conclusion that MincSathre and C. P. ChenG. Wang’s inequality are extended and refined. ..."
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We studied the monotonicity and Convexity properties of the new functions involving the gamma function, and get the general conclusion that MincSathre and C. P. ChenG. Wang’s inequality are extended and refined.
A monotonicity property of the Γfunction
"... A monotonicity property of the Γfunction ..."
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The authors were partly supported by the DFG..
, 2002
"... ABSTRACT. It is shown that the function x ↦ → 1+ 1 ..."
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"... It is shown that the function x ↦ → 1 + 1 x ln Γ(x + 1) − ln(x + 1) is strictly completely monotone on (−1, ∞) and tends to one as x → −1, to zero as x → ∞. This property is derived from a suitable integral representation of ln Γ(x + 1). MSC 2000: 33B15 ..."
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It is shown that the function x ↦ → 1 + 1 x ln Γ(x + 1) − ln(x + 1) is strictly completely monotone on (−1, ∞) and tends to one as x → −1, to zero as x → ∞. This property is derived from a suitable integral representation of ln Γ(x + 1). MSC 2000: 33B15