MathLabs

第2题

设 n≥2 n\ge2 be a fixed 整数. (a) Find least 常数 C C 使得 对所有 nonnegative 实数 x1,…,xn x_1,\dots,x_n , ∑1≤i<j≤nxixj(xi2+xj2)≤C(∑i=1nxi)4\sum_{1\le i<j\le n}x_ix_j(x_i^2+x_j^2)\le C\left(\sum_{i=1}^n x_i\right)^4. (b) 确定 when equality occurs 对于th是value 的 C C .
第 5/5 步：译文：Characterize equality
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通俗地说

Both inequalities 必须be sharp simultaneously.

xi=xj>0,xℓ=0 (ℓ≠i,j)x_i=x_j>0,\quad x_\ell=0\ (\ell\ne i,j)
详细分析

F或a nonzero tuple, equality 中 xi2+xj2≤Q x_i^2+x_j^2\le Q requires th在所有variables except two members 的contributing pair vanish. Equality 中AM–GM requires Q=2R Q=2R , which 对于two remaining variables 是 xi=xj x_i=x_j . 反之, th在configurati上gives equality. (The all-zero tuple 是trivial equality case.)