Weighted Gehring And Muckenhoupt Classes And Some Inclusion Properties With Norm Inequalities

In this paper, we will prove some fundamental properties of the power mean operator H(lambda)w(p) of order p, which is defined byH(lambda)w(p) (chi) = 1/Lambda (chi)integral(x)(0) lambda (s) w(p) (s) ds, for p is an element of R+,where lambda and w are nonnegative functions and Lambda (chi) = f(0)(chi) lambda (s) ds. Then by using these properties we will establish some norm inequalities of the generalized Muckenhoupt and Gehring weights and prove some fundamental relations between them.