Further Inequalities for the Gamma Function

Laforgia, Andrea

doi:10.2307/2007604

Cited by 19 publications

(16 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then

X - k

, which has the same distribution as

Urn (1, k m, k, n - m)

, is distributed as a binomial random variable with parameters

n - m

and

Z

. Note that

P r [Z < 3 / (m f (n))] = \frac{Γ (m + 1 / k)}{Γ (m) Γ (1 / k)} \int_{0}^{3 / (m f (n))} x^{1 / k - 1} {(1 - x)}^{m - 1} d x

< \frac{m^{1 / k}}{Γ (1 / k)} \int_{0}^{3 / (m f (n))} x^{1 / k - 1} d x = \frac{3^{1 / k} k}{Γ (1 / k) f {(n)}^{1 / k}} = o (1),

where we have used the fact

Γ (m + 1 / k) < Γ (m) m^{1 / k}

which follows from , inequality (2.2)]. On the other hand, the Chernoff bound (see, e.g., , Theorem 4.2]) gives

P

…”

Section: Proof Of Theoremmentioning

confidence: 99%

Randomized rumor spreading in poorly connected small-world networks

Mehrabian

Pourmiri

2015

Random Struct. Alg.

View full text Add to dashboard Cite

ABSTRACT:The Push-Pull protocol is a well-studied round-robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread it to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze the behavior of this protocol on random k-trees, a class of power law graphs, which are small-world and have large clustering coefficients, built as follows: initially we have a k-clique. In every step a new node is born, a random k-clique of the current graph is chosen, and the new node is joined to all nodes of the k-clique. When k 2 is fixed, we show that if initially a random node is aware of the rumor, then with probability 1 − o(1) after O (log n) 1+2/k · log log n · f (n) rounds the rumor propagates to n − o(n) nodes, where n is the number of nodes and f (n) is any slowly growing function. Since these graphs have polynomially small conductance, vertex expansion O(1/n) and constant treewidth, these results demonstrate that Push-Pull can be efficient even on poorly connected networks.On the negative side, we prove that with probability 1 − o(1) the protocol needs at leastn) rounds to inform all nodes. This exponential dichotomy between time required for informing almost all and all nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound successfully carries over to a closely related class of graphs, the random k-Apollonian networks, for which we prove an upper bound of O ((log n) c k · log log n · f (n)) rounds for informing n − o(n) nodes with probability 1 − o(1) when k 3 is fixed. Here, c k = (k 2 − 3)/(k − 1) 2 < 1 + 2/k.

show abstract

“…Then

X - k

, which has the same distribution as

Urn (1, k m, k, n - m)

, is distributed as a binomial random variable with parameters

n - m

and

Z

. Note that

P r [Z < 3 / (m f (n))] = \frac{Γ (m + 1 / k)}{Γ (m) Γ (1 / k)} \int_{0}^{3 / (m f (n))} x^{1 / k - 1} {(1 - x)}^{m - 1} d x

< \frac{m^{1 / k}}{Γ (1 / k)} \int_{0}^{3 / (m f (n))} x^{1 / k - 1} d x = \frac{3^{1 / k} k}{Γ (1 / k) f {(n)}^{1 / k}} = o (1),

where we have used the fact

Γ (m + 1 / k) < Γ (m) m^{1 / k}

which follows from , inequality (2.2)]. On the other hand, the Chernoff bound (see, e.g., , Theorem 4.2]) gives

P

…”

Section: Proof Of Theoremmentioning

confidence: 99%

Randomized rumor spreading in poorly connected small-world networks

Mehrabian

Pourmiri

2015

Random Struct. Alg.

View full text Add to dashboard Cite

show abstract

“…The right-hand side inequality in (22) is the same as (7) essentially. It is noted that a more strengthened conclusion than the right-hand side inequality in (22) has been established in [12, p. 250] and [44, Proposition 4]: Let s and t be two real numbers and = min{s, t}. Then the function…”

Section: 8mentioning

confidence: 99%

“…In the left-hand side inequality of (22), substituting x by x + s and y by x + t for two real numbers s and t and x ∈ (− min{s, t}, ∞) leads to…”

Section: 8mentioning

confidence: 99%

See 1 more Smart Citation

A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality

Guo

2008

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…For large γ, using the Gaussian Q-function approximation for small arguments Q(x) ≤ (1/2) exp (−x 2 /2), and the inequality for the ratio of gamma functions [7], so that Γ (x + 1/2)/Γ (x + 1) < (x + 1/4) −1/2 , a very simple upper bound on (14) results in…”

Section: Modified Space-time Block Codementioning

confidence: 99%

Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence

García-Zambrana

2007

IEEE Commun. Lett.

View full text Add to dashboard Cite

In this letter, error rate performance for space-time block coding (STBC) in free-space optical (FSO) communication systems with direct detection operating over strong atmospheric turbulence channels is analyzed. Based on the modification of the Alamouti code presented by Simon and Vilnrotter, a generalized approach is adopted to consider space-time coded on-off keying (OOK) formats with any pulse shape and reduced duty cycle, allowing the increase of the peak-to-average optical power ratio (PAOPR) and, hence, a relevant improvement in error rate performance. Simulation results are further demonstrated to confirm the analytical results.

show abstract

Further Inequalities for the Gamma Function

Cited by 19 publications

References 3 publications

Randomized rumor spreading in poorly connected small-world networks

Randomized rumor spreading in poorly connected small-world networks

A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality

Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence

Contact Info

Product

Resources

About