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We present the best constant and the existence of extremal functions for anImproved Hardy-Sobolev inequality. We prove that, under a propertransformation, this inequality is equivalent to the Sobolev inequality in$\mathbb{R}^N$. We also discuss the connection of the related functional spacesand as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Ourstarting point is the existence of a minimizer for the Bliss' inequality andthe indirect dependence of the Hardy inequality at the origin.

The Best Constant, the Nonexistence of Extremal Functions and Related Results for an Improved Hardy-Sobolev Inequality