Properties of a Generalized Class of Weights Satisfying Reverse H?lder’s Inequality

摘要

In this paper, we will prove some fundamental properties of the discrete power mean operator of order , where is a nonnegative discrete weight defined on the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, we will study the structure of the generalized discrete class of weights that satisfy the reverse H?lder inequality for positive real numbers , , and such that and . For applications, we will prove some self-improving properties of weights from and derive the self improving properties of the discrete Gehring weights as a special case. The paper ends by a conjecture with an illustrative sharp example.