鉴于近年来发表的一些文章中关于不等式的对称与轮换对称这两个概念及性质运用模糊,往往导致错误,笔者就此问题作初步的探讨,供大家参考。 一、关于不等式对称与轮换对称的定义 在一个不等式中,若把其中任何两个字母a_i和a_j(i,i=1,2,…,n,且i≠j)对调位置后,这个不等式不变(如①:a/(b+c)+b/(c+a)+c/(a+b)≥3/2,其中a,b,c>0),我们便称此不等式是关于a_1、a_2、…、a_n对称的,如果把不等式中的卞母a_1、a_2、…;a_n按一定顺序顺次替换(如将a_1换成a_2,a_2换成a_3,…,a_(n-1)换成a_n,a_n换成a_1)后不等式不变(如②:(b~2-c~2)/(a+b)+(c~2-a~2)/(b+c)+(a~2-b~2)/(c+a)≥0,其中a,b,c∈R~+),我们便称此不等式是关于a_1、a_2、…、a_n轮换对 In view of the ambiguity in the concepts and nature of inequality symmetry and rotational symmetry in some articles published in recent years, which often leads to errors, the author makes a preliminary discussion on this issue for your reference. First, on the definition of inequality symmetry and rotation symmetry In an inequality, if any two of the letters a_i and a_j (i,i=1,2,...,n, and i≠j) are reversed, this inequality does not Change (eg, 1:a/(b+c)+b/(c+a)+c/(a+b)≥3/2, where a,b,c>0), we call this inequality If a_1, a_2, ..., a_n are symmetric, if the mothers a_1, a_2, ..., a_n in the inequality are sequentially replaced in a certain order (for example, a_1 is replaced by a_2, a_2 is replaced by a_3, ..., a_(n-1) ) After replacing a_n, a_n with a_1, the inequality does not change (eg, 2:(b~2-c~2)/(a+b)+(c~2-a~2)/(b+c)+ (a~2-b~2)/(c+a)≥0, where a,b,c∈R~+), we call this inequality about a_1, a_2, ..., a_n rotation pair