  The geometric discord $\mathcal{D}$ of a state is a measure of the
quantumness of the state and the negativity $\mathcal{N}$ is a measure of the
entanglement of a state. It was proved by D. Girolami and G. Adesso that for
states on $\mathbb{C}^2\otimes\mathbb{C}^2$, the geometric discord is always
greater than or equal to the square of the negativity and conjectured that this
holds in general. S. Rana and P. Parashar showed that this relation does not
hold for all states on $\mathbb{C}^2\otimes\mathbb{C}^n$ for $n>2$. We provide
several analytic families of states on $\mathbb{C}^2\otimes\mathbb{C}^3$
violating this relation. Certain upper and lower bounds for
$\mathcal{N}^2-\mathcal{D}$ are obtained for states on
$\mathbb{C}^m\otimes\mathbb{C}^n$ for any $m, n\in\mathbb{N}$.
