Extension and Application to the Nature of the Average Inequality

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