Logarithmically Concave Sequence

A finite sequence of real numbers  is said to be logarithmically concave (or log-concave) if

holds for every  with .

A logarithmically concave sequence of positive numbers is also unimodal.

If  and  are two positive log-concave sequences of the same length, then  is also log-concave. In addition, if the polynomial  has all its zeros real, then the sequence  is log-concave (Levit and Mandrescu 2005).

An example of a logarithmically concave sequence is the sequence of binomial coefficients  for fixed  and .

REFERENCES:

Levit, V. E. and Mandrescu, E. "The Independence Polynomial of a Graph--A Survey." In Proceedings of the 1st International Conference on Algebraic Informatics. Held in Thessaloniki, October 20-23, 2005 (Ed. S. Bozapalidis, A. Kalampakas, and G. Rahonis). Thessaloniki, Greece: Aristotle Univ., pp. 233-254, 2005.