Type of sequence of numbers
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.
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.
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