Context: there are $2$ goods with prices $P_1$ and $P_2$ and the decision maker has the utility function $U(C_1,C_2)$. Denote $U_j=\frac{\partial U(C_1,C_2)}{\partial C_j}$ for $j\in\{1,2\}$. A good $j$ is normal if its income elasticity of demand is positive.
The author then claims that a sufficient condition that good $j$ is normal is $$ (U_i/U_j)U_{jj}-U_{ij}<0. $$ Could you please help me understanding why this condition is indeed sufficient for good $j$ to be normal? Thank you very much!
More context: this comes from the lecture materials of a microeconomics class. If you need to, please make the standard assumptions concerning $U$: $U_i>0$, $U_{ii}<0$.