均值不等式的性质推广及应用

Extension and Application to the Nature of the Average Inequality

不等式在数学中占有重要的地位.不等式的证明经常用到算术平均数、几何平均数、调和平均数之间的关系.本文着重讲述了这几种均值不等式之间的关系并加以推广,以及对均值不等式在指数方面作了推广,并且将"n个正数的算数平均数大于等于几何平均数"这一重要不等式推广到"加权算术平均值的函数与函数值的加权算术平均值之间的关系",继而得出结论"n个正数的加权算术平均数不小于它们的加权几何平均数",同时向矩阵方面加以推广.

Inequality plays an important role in mathematics.The certification of inequality have frequently used the relations between arithmetic mean,geometric mean and harmonic mean.In this paper,an account of the relationship between several of inequality were given,and the average index of inequality was promoted.It was also promoted that the important inequality ＂the arithmetic mean of n positive number is greater than or equal the geometric mean＂ to ＂the relations between weighted arithmetic mean of the function and function value and the weighted arithmetic mean＂.It was concluded that the weighted arithmetic average of n positive number is not less than their weighted geometric mean.