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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 38 (1 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) =
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.
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 15 (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 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
Inequalities for averages of convex and superquadratic functions
 J. Inequal. in Pure & Appl. Math
"... ABSTRACT. We consider the averages An(f) = 1/(n − 1) ∑n−1 r=1 f(r/n) and Bn(f) = 1/(n + 1) ∑n r=0 f(r/n). If f is convex, then An(f) increases with n and Bn(f) decreases. For the class of functions called superquadratic, a lower bound is given for the successive differences in these sequences, in t ..."
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Cited by 6 (6 self)
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ABSTRACT. We consider the averages An(f) = 1/(n − 1) ∑n−1 r=1 f(r/n) and Bn(f) = 1/(n + 1) ∑n r=0 f(r/n). If f is convex, then An(f) increases with n and Bn(f) decreases. For the class of functions called superquadratic, a lower bound is given for the successive differences in these sequences, in the form of a convex combination of functional values, in all cases at least f(1/3n). Generalizations are formulated in which r/n is replaced by ar/an and 1/n by 1/cn. Inequalities are derived involving the sum ∑n r=1 (2r − 1)p.
Note On An Inequality Involving ...
, 1995
"... . We prove: If G(n) = (n!) 1=n denotes the geometric mean of the first n positive integers, then 1 e 2 ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) holds for all n 2. The lower bound 1 e 2 is best possible. In 1964 H. Minc and L. Sathre [2] published several remarkable inequalities involving th ..."
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. We prove: If G(n) = (n!) 1=n denotes the geometric mean of the first n positive integers, then 1 e 2 ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) holds for all n 2. The lower bound 1 e 2 is best possible. In 1964 H. Minc and L. Sathre [2] published several remarkable inequalities involving the geometric mean of the first n positive integers. Their main result states: If G(n) = (n!) 1=n , then (1) 1 ! n G(n + 1) G(n) \Gamma (n \Gamma 1) G(n) G(n \Gamma 1) holds for all integers n 2. The lower bound 1 is best possible. Recently, the author [1] proved the following refinement of (1): If n 2, then (2) 1 ! 1 + G(n) G(n \Gamma 1) \Gamma G(n + 1) G(n) ! n G(n + 1) G(n) \Gamma (n \Gamma 1) G(n) G(n \Gamma 1) : The lefthand inequality of (2), written as (3) 0 ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) (n 2); leads to the question: What is the greatest real number c (which is independent of n) such that c ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) holds for all n 2? It is...
Acta Math. Univ. Comenianae Vol. LXIV, 2(1995), pp. 283285
"... . We prove: If G(n) = (n!) 1=n denotes the geometric mean of the first n positive integers, then 1 e 2 ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) holds for all n 2. The lower bound 1 e 2 is best possible. In 1964 H. Minc and L. Sathre [2] published several remarkable inequalities involving th ..."
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. We prove: If G(n) = (n!) 1=n denotes the geometric mean of the first n positive integers, then 1 e 2 ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) holds for all n 2. The lower bound 1 e 2 is best possible. In 1964 H. Minc and L. Sathre [2] published several remarkable inequalities involving the geometric mean of the first n positive integers. Their main result states: If G(n) = (n!) 1=n , then (1) 1 ! n G(n + 1) G(n) \Gamma (n \Gamma 1) G(n) G(n \Gamma 1) holds for all integers n 2. The lower bound 1 is best possible. Recently, the author [1] proved the following refinement of (1): If n 2, then (2) 1 ! 1 + G(n) G(n \Gamma 1) \Gamma G(n + 1) G(n) ! n G(n + 1) G(n) \Gamma (n \Gamma 1) G(n) G(n \Gamma 1) : The lefthand inequality of (2), written as (3) 0 ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) (n 2); leads to the question: What is the greatest real number c (which is independent of n) such that c ! (G(n)) 2 \Gamma G(n \Gamma 1)G(n + 1) holds for all n 2? It is...