On weighted discrete Hardy's inequality for negative power numbers

Abstract

In this paper we consider the weighted discrete Hardy’s inequality for different real power numbers 0 = r < 1 and obtain some new refinements of its finite sections. For r < −1 our results improve those previously given by Nguyen et al. in [19, 20]. Moreover, we prove that the constant factors involved in the right-hand sides of the obtained inequalities are the best possible, that is, they cannot be replaced with any smaller constant. Mathematics subject classification (2000): 26D15.