MathDB

Find the maximal constant

M

, such that for arbitrary integer

n\geq 3,

there exist two sequences of positive real number

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

and

b_{1},b_{2},\cdots,b_{n},

satisfying (1): \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1; (2): a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M.

Problem Statement