对一类三元n次不等式的证明

• 作 者: 杨学枝
• 机 构: 福建省福州第二十四中学,
• 出 处: 数学通报
• 发表日期: 2017年11期

以下四个定理可用于证明一类三元n次不等式.文中“∑”与“∏”分别为三元数的循环和与循环积.定理一若(1)λ,u≥v≥0,w≥0;(2)(λ-1)(λ-w)≤0,(u-1)(u-w)≤0(或λ,v≥u≥0,w≥0,(λ-1)(λ-w)≤0,(v-1)(v-w)≤0;或u,v≥λ≥0,w≥0,(u-1)(u-w)≤0,(v-1)(v-w)≤0);(3)(λ-1)(λ-w)+(u-1)·(u-w)+(v-1)(v-w)≤0;(4)λ+u+v≤1+2w.则对于n∈N,n≥2,有

The following four theorems can be used to prove a class of ternary n-degree inequalities. In the text, “∑” and “∏” are ternary cycles and circulatory products respectively. Theorem one if (1) λ, u ≥ v ≥ 0, w ≥ 0; (2) (λ-1) (λ-w) ≤ 0, (u-1) (uw) ≤ 0 (or λ, v ≥ u ≥ 0, w ≥ 0, (λ- 1) (λ-w) ≤ 0, (v-1) (vw) ≤ 0; or u, v ≥ λ ≥ 0, w ≥ 0, (u-1) (uw) ≤ 0, (v-1) (vw) ≤ 0); (3) (λ-1) (λ-w) + (u-1) · (uw) + (v-1) (vw) ≤ 0; (4) λ + u + v ≤1+2w. Then for n∈N, n≥2, there are