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A geometrical proof of a new inequality for the gamma function
 J. Ineq. Pure Appl. Math
"... ABSTRACT. Using the inclusions between the unit balls for the pnorms, we obtain a new inequality for the gamma function. ..."
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ABSTRACT. Using the inclusions between the unit balls for the pnorms, we obtain a new inequality for the gamma function.
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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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.
Logarithmically completely monotonic functions relating to the gamma function
 J. Math. Anal. Appl
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Some classes of completely monotonic functions
, 2002
"... Abstract. A function f: (0, ∞) → R is said to be completely monotonic if (−1) n f (n) (x) ≥ 0 for all x> 0 and n = 0, 1, 2,.... In this paper we present several new classes of completely monotonic functions. Our functions have in common that they are defined in terms of the classical gamma, dig ..."
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Abstract. A function f: (0, ∞) → R is said to be completely monotonic if (−1) n f (n) (x) ≥ 0 for all x> 0 and n = 0, 1, 2,.... In this paper we present several new classes of completely monotonic functions. Our functions have in common that they are defined in terms of the classical gamma, digamma, and polygamma functions. Moreover, we apply one of our monotonicity theorems to prove a new inequality for prime numbers. Some of the given results extend and complement theorems due to Bustoz & Ismail, Clark & Ismail, and other researchers. Key words. Complete monotonicity, gamma, digamma, and polygamma functions, prime numbers, inequalities. 2000 Mathematics Subject Classification. Primary 11A41, 26A48, 33B15; Secondary 26D15. 1.
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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Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x
Some new inequalities for gamma and polygamma functions
 Art. 103; Available online at http://jipam.vu.edu.au/article.php? sid=577. F. QI AND B.N. GUO
"... ABSTRACT. In this paper we derive some new inequalities involving the gamma function Γ, polygamma functions ψ = Γ ′ /Γ and ψ ′. We also obtained two new sequences converging to EulerMascheroni constant γ very quickly. ..."
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ABSTRACT. In this paper we derive some new inequalities involving the gamma function Γ, polygamma functions ψ = Γ ′ /Γ and ψ ′. We also obtained two new sequences converging to EulerMascheroni constant γ very quickly.
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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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 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 ..."
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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.
Stochastic Complexities of Gaussian Mixtures In Variational Bayesian Approximation
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2006
"... Bayesian learning has been widely used and proved to be effective in many data modeling problems. However, ..."
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Bayesian learning has been widely used and proved to be effective in many data modeling problems. However,