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In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property ( backward propagation property ) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponents as well as the new constants of the new classes. As an application, we establish the self-improving property ( forward propagation property ) of the discrete Gehring class.

On discrete weighted Hardy type inequalities and properties of weighted discrete Muckenhoupt classes