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When studying the weighted Hardy-Rellich inequality in L² with the full gradient replaced by the radial derivative, the best constant becomes trivially larger or equal than in the first situation. Our contribution is to determine the new sharp constant and to show that for some part of the weights is strictly larger than before. In some cases, we emphasize that the extremals functions of the sharp constant are not radially symmetric. [ABSTRACT FROM AUTHOR]

In this survey we give a compact presentation of well-known functional inequalities of Hardy and Rellich type in the L2 setting. In addition, we give some insights into the proofs by using basic tools with emphasis on the particularities of a more general approach which is the method of super-solutions. [ABSTRACT FROM AUTHOR]