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

Abstract In this paper, we will prove some fundamental properties of the power mean operator $$ \mathcal{M}_{p}g(t)= \biggl( \frac{1}{\Upsilon(t)} \int _{0}^{t} \lambda (s)g^{p} ( s ) \,ds \biggr) ^{1/p},\quad\text{for }t\in \mathbb{I}\subseteq \mathbb{R}_{+}, M p g ( t ) = 1 ϒ ∫ 0 λ s d / , for ∈ I ⊆ R + order p and establish lower upper bounds compositions operators different powers, where g , λ are a nonnegative real valued functions defined on $\mathbb{I}$ $\Upsilon(t)=\int _{0}^{t}\lambda \,ds$ . Next, study structure generalized class $\mathcal{U}_{p}^{q}(B)$ U q B weights that satisfy reverse Hölder inequality \mathcal{M}_{q}u\leq B\mathcal{M}_{p}u, u ≤ for $p< q$ < $p.q\neq 0$ . ≠ $B>1$ > is constant. For applications, self-improving in derive self improving weighted Muckenhoupt Gehring classes.