Structure of a generalized class of weights satisfy weighted reverse Hölder's inequality.

In this paper, we will prove some fundamental properties of the power mean operator M p g (t) = (1 ϒ (t) ∫ 0 t λ (s) g p (s) d s) 1 / p , for t ∈ I ⊆ R + , of order p and establish some lower and upper bounds of the compositions of operators of different powers, where g, λ are a nonnegative real valued functions defined on I and ϒ (t) = ∫ 0 t λ (s) d s . Next, we will study the structure of the generalized class U p q (B) of weights that satisfy the reverse Hölder inequality M q u ≤ B M p u , for some p < q , p. q ≠ 0 , and B > 1 is a constant. For applications, we will prove some self-improving properties of weights in the class U p q (B) and derive the self improving properties of the weighted Muckenhoupt and Gehring classes. [ABSTRACT FROM AUTHOR]