logarithmically concave sequence

{{Short description|Type of sequence of numbers}}

File:Pascal's_triangle_5.svg

In mathematics, a sequence {{math|a}} = {{math| (a₀, a₁, ..., a_n)}} of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if {{math|a_i² ≥ a_i−1a_i+1}} holds for {{math|0 < i < n }}.

Remark: some authors (explicitly or not) add two further conditions in the definition of log-concave sequences:

• {{math|a}} is non-negative
• {{math|a}} has no internal zeros; in other words, the support of {{math|a}} is an interval of {{math|Z}}.

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 {{Cite book|last=Brenti|first=Francesco|url=|title=Unimodal, log-concave and Pólya frequency sequences in combinatorics|year=1989|publisher=American Mathematical Society|isbn=978-1-4704-0836-7|location=Providence, R.I.|oclc=851087212}} for a discussion on the two notions. For instance, the sequence {{math|(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.