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Logarithmically Concave Sequence

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A logarithmically concave sequence (or log-concave sequence) of real numbers {a_k}_(k=1)^n satisfies

a_i^2>=a_(i-1)a_(i+1)

for every a_i with 2<=i<=n-1.

For n>=1, a logarithmically concave sequence of positive numbers is also unimodal.

If {a_i} and {b_i} are two positive log-concave sequences of the same length, then {a_ib_i} is also log-concave. In addition, if the polynomial sum_(i=0)^(n)p_ix^i has real coefficients and all its zeros are real, then the sequence {p_i/(n; i)} is log-concave (Levit and Mandrescu 2005, p. 237).

An example of a logarithmically concave sequence is the sequence of binomial coefficients (n; k) for fixed n and 0<=k<=n.

See also

Infinitely Logarithmically Concave Sequence, Logarithmically Concave Function, Logarithmically Concave Polynomial, Logarithmically Convex Sequence, Real-Rooted Polynomial, Ultra Log-Concave Sequence, Unimodal Sequence

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References

Levit, V. E. and Mandrescu, E. "The Independence Polynomial of a Graph--A Survey." In Proc. 1st International Conference on Algebraic Informatics (Ed. S. Bozapalidis, A. Kalampakas, and G. Rahonis). Thessaloniki, Greece: Aristotle Univ., pp. 231-252, 2005.

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Logarithmically Concave Sequence

Cite this as:

Weisstein, Eric W. "Logarithmically Concave Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html

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