构造矩阵法在不等式证明中的应用

Application of Construction Matrix Method to Proof of Inequality

根据欲证不等式本身的特征构造或设计一个具有一定规律和模式的非负实数矩阵,利用矩阵中元素之间的算术平均值与几何平均值的某些关系,证明一些不等式,并推广已有文献的一些结论。这一非常规特殊方法比常规的一般方法更加简捷、更加有效,往往会起到事半功倍作用。

A non-negative real number matrix with certain rules and patterns is constructed or designed according to the characteristics of the inequality to be proved. Some inequalities are proved by using some relations between the arithmetic mean and the geometric mean of the elements in the matrix, and some conclusions in the literature are generalized. This unconventional special method is simpler and more effective than the conventional general method, and it often works twice as well with half the effort.