Twice Means And Hardy Inequality

The main object of this thesis is devoted to improve the classic Hardy inequality by the means of average.We get a integral core " In (x/t) " by solving the problem of two times integral means.Then there comes the definition of twice means.Moreover we extend it in the following two aspects: On the one hand,basing on the twice means,we define twice means of arbitrary order and proof a series of mixed-means inequalities.On the other hand,we get the Hardy inequality and the Carleman inequality with twice means.Furthermore,we extend the results into the Nth means.