Discrete Hardy's type inequalities and structure of discrete class of weights satisfy reverse Hölder's inequality

In this paper, we will prove a new discrete weighted Hardy’s type inequality with different powers. Next, we will apply this inequality to prove that the forward and backward propagation properties (self-improving properties) for the general discrete class Bp,q of weights that satisfy the reverse Hölder inequality hold. As special cases, we will deduce the self-improving properties of discrete Gehring and Muckenhoupt weights. An example is considered for illustrations. Mathematics subject classification (2010): 26D07, 40D25, 42C10 43A55,46A35, 46B15.