MathDB

Let

0\leq a_1\leq a_2\leq \cdots\leq a_{n-1}\leq a_n

and

a_1+a_2+\cdots+a_n=1.

Prove that: For any non-negative numbers

x_1,x_2,\cdots,x_n ; y_1, y_2,\cdots, y_n

, have

\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right) \left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.

Problem Statement