MathDB

Let

\, a_{0}, a_{1}, a_{2},\ldots\,

be a sequence of positive real numbers satisfying \, a_{i\minus{}1}a_{i\plus{}1}\leq a_{i}^{2}\, for i \equal{} 1,2,3,\ldots\; . (Such a sequence is said to be log concave.) Show that for each

\, n > 1,

\frac{a_{0}\plus{}\cdots\plus{}a_{n}}{n\plus{}1}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n\minus{}1}}{n\minus{}1}\geq\frac{a_{0}\plus{}\cdots\plus{}a_{n\minus{}1}}{n}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n}}{n}.

Problem Statement