Logarithmically concave sequence

In mathematics, a sequence {{{1}}} of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a_i² ≥ a_i−1a_i+1 holds for 0 < i < n .

Remark: some authors (explicitely or not) add two further hypotheses in the definition of log-concaves sequences:

• a is non-negative
• a has no internal zeros; in other words, the support of a is a connected interval of Z.

For instance, the sequence (1,1,0,0,1) checks the inequalities but not the internal zeros condition.

Examples of log-concave sequences are given by the binomial coefficients along any row of Pascal's triangle.

References

• Stanley, R. P. (December 1989). "Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry". Annals of the New York Academy of Sciences. 576: 500–535. doi:10.1111/j.1749-6632.1989.tb16434.x.

See also

• Unimodality
• Logarithmically concave function
• Logarithmically concave measure