Logarithmically concave sequence: Difference between revisions

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In [[mathematics]], a sequence {{math|''a''}} = {{math| (''a''₀, ''a''₁, ..., ''a''_''n'')}} of nonnegative real numbers is called a '''logarithmically concave sequence''', or a '''log-concave sequence''' for short, if {{math|''a''_''i''² ≥ ''a''_''i''−1''a''_''i''+1}} holds for {{math|0 < ''i'' < ''n'' }}.

'''Remark:''' some authors (explicitly or not) add two further hypothesesconditions 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 hypothesesconditions mirror the ones required for [[Logarithmically_concave_function|log-concave functions]].

Sequences that fulfill the three conditions are also called '''Pòlya Frequency sequences of order 2''' ('''PF₂''' sequences). Refer to chapter 2 of <ref name="brenti">Brenti, F. (1989). Unimodal Log-Concave and Pòlya Frequency Sequences in Combinatorics. American Mathematical Society.</ref> for a discussion on the two notions.

For instance, the sequence {{math|(1,1,0,0,1)}} checkssatisfies the concavity inequalities but not the internal zeros condition.

Examples of log-concave sequences are given by the [[binomial coefficient]]s along any row of [[Pascal's triangle]] and the [[Newton's inequalities|elementary symmetric means]] of a finite sequence of real numbers.