Results 1  10
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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 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 completely monotonic function involving divided differences of psi and polygamma functions and an application
 RGMIA Res. Rep. Coll
"... Abstract. A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw’s double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of t ..."
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Cited by 17 (13 self)
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Abstract. A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw’s double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw’s double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in R n−1 and R n respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized. 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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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.
On some inequalities for the incomplete gamma function
 Math. Comp
, 1997
"... Abstract. Let p � = 1 be a positive real number. We determine all real numbers α = α(p) andβ=β(p) such that the inequalities [1 − e −βxp] 1/p � 1 ..."
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Cited by 14 (0 self)
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Abstract. Let p � = 1 be a positive real number. We determine all real numbers α = α(p) andβ=β(p) such that the inequalities [1 − e −βxp] 1/p � 1
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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Cited by 5 (5 self)
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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
Rayleigh Flat Fading Channels ’ Capacity
"... In this paper, we consider singleuser transmission over a Rayleigh flat fading channel, in which the Channel State Information (CSI) is known by the receiver only. Subject to an average transmit power constraint, we study the capacity of an Additive White Gaussian Noise (AWGN) channel with Rayleigh ..."
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Cited by 4 (0 self)
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In this paper, we consider singleuser transmission over a Rayleigh flat fading channel, in which the Channel State Information (CSI) is known by the receiver only. Subject to an average transmit power constraint, we study the capacity of an Additive White Gaussian Noise (AWGN) channel with Rayleigh fading. Under an independently identically distributed fading assumption, lower and upper bounds of the channel capacity are given and proved and they are compared to the capacity results numerically computed. Besides, an approximation result of such channel capacity is proposed, and by conducting numerical comparison it is shown that our suggested approximation result has a better performance in approximating Rayleigh fading channels capacity than the bounds given above. In addition, the channel capacity with outage probability is discussed and compared with different outage probabilities.
AN ALTERNATIVE PROOF OF ELEZOVIĆGIORDANOPEČARIĆ’S THEOREM
, 903
"... Abstract. In the present note, an alternative proof is supplied for Theorem 1 ..."
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Cited by 3 (3 self)
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Abstract. In the present note, an alternative proof is supplied for Theorem 1
Correct ordering in the Zipf–Poisson ensemble
, 2010
"... We consider a Zipf–Poisson ensemble in which Xi ∼ Poi(Ni −α) for α> 1 and N> 0 and integers i ≥ 1. As N → ∞ the first n ′ (N) random variables have their proper order X1> X2> · · ·> Xn ′ relative to each other, with probability tending to 1 for n ′ up to (AN / log(N)) 1/(α+2) for an explicit cons ..."
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We consider a Zipf–Poisson ensemble in which Xi ∼ Poi(Ni −α) for α> 1 and N> 0 and integers i ≥ 1. As N → ∞ the first n ′ (N) random variables have their proper order X1> X2> · · ·> Xn ′ relative to each other, with probability tending to 1 for n ′ up to (AN / log(N)) 1/(α+2) for an explicit constant A(α) ≥ 3/4. The rate N 1/(α+2) cannot be achieved. The ordering of the first n ′ (N) entities does not preclude Xm> Xn ′ for some interloping m> n ′. The first n ′ ′ random variables are correctly ordered exclusive of any interlopers, with probability tending to 1 if n ′ ′ ≤ (BN / log(N)) 1/(α+2) for B < A. For a Zipf–Poisson model of the British National Corpus, which has a total word count of 100,000,000, our result estimates that the 72 words with the highest counts are properly ordered. 1
pe xp Γ(m+1) Study of a Class of Regularizations of 1/x  using Gaussian Integrals
, 1999
"... This paper presents a comprehensive study of the functions V p m(x) = x (tp − x p) m e −tp dt for x> 0, m> −1 and p> 0. For large x these functions approximate x1−p. The case p = 2 is of particular (x) ≈ 1/x can be regarded as importance because the functions V 2 m onedimensional regularizations o ..."
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This paper presents a comprehensive study of the functions V p m(x) = x (tp − x p) m e −tp dt for x> 0, m> −1 and p> 0. For large x these functions approximate x1−p. The case p = 2 is of particular (x) ≈ 1/x can be regarded as importance because the functions V 2 m onedimensional regularizations of the Coulomb potential 1/x  which are finite at the origin for m> −1 2.