Logarithmically concave sequence

In mathematics, a sequence $a$ = $(a_{0}, a_{1}, ..., a_{n})$ of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if $a_{i}^{2} ≥ a_{i−1}a_{i+1}$ holds for $0 < i < n$.

Remark: some authors (explicitly or not) add two further conditions in the definition of log-concave sequences: These conditions mirror the ones required for log-concave functions.
 * $a$ is non-negative
 * $a$ has no internal zeros; in other words, the support of $a$ is an interval of $Z$.

Sequences that fulfill the three conditions are also called Pólya Frequency sequences of order 2 (PF2 sequences). Refer to chapter 2 of for a discussion on the two notions. For instance, the sequence $(1,1,0,0,1)$ satisfies the concavity 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 and the elementary symmetric means of a finite sequence of real numbers.