  We give necessary and sufficient conditions on a pair of positive radial
functions $V$ and $W$ on a ball $B$ of radius $R$ in $R^{n}$, $n \geq 1$, so
that the following inequalities hold \begin{equation*} \label{two}
\hbox{$\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x) u^{2}dx+b\int_{\partial
B}u^2 ds$ for all u $\in H^1(B)$,} \end{equation*} and \begin{equation*}
\label{two} \hbox{$\int_{B}V(x)|\Delta u |^{2}dx \geq \int_{B} W(x)|\nabla
u|^{2}dx+b\int_{\partial B}|\nabla u|^2 ds$ for all u $\in H^2(B)$.}
\end{equation*} Then we present various classes of optimal weighted
Hardy-Rellich inequalities on $H^{2}\cap H^{1}_{0}$. The proofs are based on
decomposition into spherical harmonics. These types inequalities are important
in the study of fourth order elliptic equations with Navier boundary condition
and systems of second order elliptic equations.
