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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.
A Logarithmically Completely monotonic Function Involving the Gamma Functions 1
"... We show that the function x → [Γ(x+1)]1/x x[Γ(x+2)] 1/(x+1) is logarithmically completely monotonic on (0, ∞). This answers a question by A.Vernescu. ..."
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Cited by 16 (12 self)
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We show that the function x → [Γ(x+1)]1/x x[Γ(x+2)] 1/(x+1) is logarithmically completely monotonic on (0, ∞). This answers a question by A.Vernescu.
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, digamm ..."
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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.
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.
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.
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
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.
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,
Inequalities Involving Gamma and Psi Functions
"... We prove that certain functions involving the gamma and qgamma function are monotone. We also prove that (x m /(x)) (m+1) is completely monotonic. We conjecture that (x m / (m\Gamma1 (x)) (m) is completely monotonic for m 2, we prove it, with help from Maple, for 2 m 16. We give a ve ..."
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We prove that certain functions involving the gamma and qgamma function are monotone. We also prove that (x m /(x)) (m+1) is completely monotonic. We conjecture that (x m / (m\Gamma1 (x)) (m) is completely monotonic for m 2, we prove it, with help from Maple, for 2 m 16. We give a very useful Maple proceedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive. Running Title: Gamma Function Inequalities Mathematics Subject Classification. Primary 33B15. Secondary 26D07, 26D10. Key words and phrases. gamma function, digamma function, inequalities, complete monotonicity. 1. Introduction. Inequalities of functions involving gamma functions have been of interest since the 1950's when inequalities by Gautchi, Erber and Kershaw were established. For references and generalizations we refer the interested reader to [5], [13], [14], [15], [16], and to Alzer's p...
Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, submitted
"... Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, clai ..."
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Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years. Contents