We investigate necessary and sufficient conditions on the weights for the
Hardy-Rellich inequalities to hold, and propose a new way to use the notion of
Bessel pair to establish the optimal Hardy-Rellich type inequalities. Our
results sharpened earlier Hardy-Rellich and Rellich type inequalities in the
literature. We also study several results about the symmetry and symmetry
breaking properties of the Rellich type and Hardy-Rellich type inequalities,
and then partially answered an open question raised by Ghoussoub and Moradifam.
Namely, we will present conditions on the weights such that the Rellich type
and Hardy-Rellich type inequalities hold for all functions if and only if the
same inequalities hold for all radial functions.Comment: 22 page