Can we divide both sides of positive semidefinite inequality elementwise by another psd matrix

Question

Assume we have positive semidefinite matrices $A,B,C$ and we have $A \circ B \ge C$. Is it correct that $A \ge C \oslash B$. In particular, for a vector $r$ and inequality, $ A \circ B \ge r \cdot r^T$. What can be said about $A$? Is $ A \ge r \cdot r^T \oslash B $ true? Where $C\ge D$ means $C-D\ge0$ is a positive semidefinite matrix,i.e. $\ge$ is in the semidefinite sense.$\odot$ is elementwise product and $\oslash$ is elementwise division.