含k-Brocard点的一类几何不等式

定义了凸N边形的k Brocard点,建立k Brocard线的长L(k)

A Class of the Geometric Inequalities Involving k-Brocard Distance

Let P be an inner point of a convex N-gon ΓN ： A1A2… ANA1（N ≥ 3）, and let di,k denote the distance from the point Ai＋k to the line PAi（i = 1,2,…,N, Ai = Aj〈=〉 i ≡ j（modN））, which is called the k-Brocard distance for P of ΓN. We have proved the following double-inequality： If P ∈ ΓN, k = N↑∩i=1∠Ai-kAiAi＋k（1 ≤ k 〈 N/2,i =1,2,…,N）, and r ≤ lnN-ln（N-1）/ln2＋2[lnN-ln（N-1）], then （1/N N↑∑↑i=1di^r, k）^1/r≤1/N coskπ/N N↑∑↑i=1｜AiAi＋k｜≤sin2kπ/2sinπ/N（1/N N↑∑↑i=1｜AiAi＋1｜^2.

A-G-H不等式的优化推广及其应用

借助于被称为降维法的新方法,建立了如下不等式:设ai>0,i=1,…,n,n≥2,A(a)1/n,H(a)=1-1-1ai,G(a)=∏n,则当且仅当实数λ≤1ai=1n∑nn时有不等式:n∑ni=1i=1i=1[H(a)]1-λ·[A(a)]λ≤G(a).作为应用,获得了一个几何不等式及一个有趣的矩阵不等式,并且推广了Carleman不等式.