A Class of the Geometric Inequalities Involving k-Brocard Distance  

作　　者：WEN Jia-jin KE Rui LUE Tao 

机构地区：[1]Department of Mathematics and Computer Science, Chengdu University, Chengdu 610106 China [2]Department of Mathematics, Guangzhou Maritime College, Guangzhou 510725, China [3]College of Mathematics, Sichuan University, Chengdu 610024, China

出　　处：《Chinese Quarterly Journal of Mathematics》2006年第2期210-219,共10页数学季刊（英文版）

基　　金：Supported by the NSF of China(10171073)

摘　　要：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.

关 键 词：convex N-gon k-Brocard distance Hoelder inequality Janous-Klamkin＇s conjecture 

分 类 号：O178[理学—数学]