MathDB

Given an integer

n\ge 2

, find the maximal constant

\lambda (n)

having the following property: if a sequence of real numbers

a_{0},a_{1},a_{2},\cdots,a_{n}

satisfies 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n}, and a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1, then (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.

Problem Statement