Inequalities for the incomplete gamma function

Natalini, Pierpaolo; Palumbo, Biagio

doi:10.7153/mia-03-08

Cited by 36 publications

(24 citation statements)

References 3 publications

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“…We use the bound Γ(a, z) < Bz a−1 e −z for a > 1, B > 1, z > B(a − 1)/(B − 1) due to [15]. Substituting a = n + 1, z = en, and B = e, we have Γ(n + 1, ν) < Γ(n + 1, en) < e(en) n e −en .…”

Section: G Proof Of Lemmamentioning

confidence: 99%

Distributed Simulation of Continuous Random Variables

Gamal

2017

IEEE Trans. Inform. Theory

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We establish the first known upper bound on the exact and Wyner's common information of n continuous random variables in terms of the dual total correlation between them (which is a generalization of mutual information). In particular, we show that when the pdf of the random variables is log-concave, there is a constant gap of n 2 log e + 9n log n between this upper bound and the dual total correlation lower bound that does not depend on the distribution. The upper bound is obtained using a computationally efficient dyadic decomposition scheme for constructing a discrete common randomness variable W from which the n random variables can be simulated in a distributed manner. We then bound the entropy of W using a new measure, which we refer to as the erosion entropy.

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Section: G Proof Of Lemmamentioning

confidence: 99%

Distributed Simulation of Continuous Random Variables

Gamal

2017

IEEE Trans. Inform. Theory

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“…These inequalities complete the results of Natalini and Palumbo [20]. We note that actually all the results stated in [20] can be formulated in terms of the Mills ratio of the gamma distribution.…”

Section: The Gamma Distributionmentioning

confidence: 53%

“…We note that actually all the results stated in [20] can be formulated in terms of the Mills ratio of the gamma distribution.…”

Section: The Gamma Distributionmentioning

confidence: 98%

Geometrically concave univariate distributions

Baricz

2010

Journal of Mathematical Analysis and Applications

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In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.

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“…for all λ > 0, and so This function is useful because Γ(a, x) ≤ Γ(a) for all x and Γ(a, x) < 2x a−1 e −x for x > a − 1 [21]. Using this function, we provide heat-kernel bounds when the volume of a ball of radius r grows exponentially for large r.…”

Section: An Upper Bound For the Heat Kernelmentioning

confidence: 99%

Upper bounds for the spectral function on homogeneous spaces via volume growth

Judge

Lyons

2019

Rev. Mat. Iberoam.

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We use spectral embeddings to give upper bounds on the spectral function of the Laplace-Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li [17] for the smallest positive eigenvalue to all eigenvalues. We also improve Li's bound itself. Our bounds translate to explicit upper bounds on the heat kernel for both compact and noncompact homogeneous spaces.

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Inequalities for the incomplete gamma function

Abstract: Abstract. We prove some monotonicity results for the incomplete gamma function,from which some inequalities for Γ(a, x) follow.Mathematics subject classification (1991): 33B15.

Cited by 36 publications

References 3 publications

Distributed Simulation of Continuous Random Variables

Distributed Simulation of Continuous Random Variables

Geometrically concave univariate distributions

Upper bounds for the spectral function on homogeneous spaces via volume growth

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