  We consider the punctured plane with volume density $|x|^\alpha$ and
perimeter density $|x|^\beta$. We show that centred balls are uniquely
isoperimetric for indices $(\alpha,\beta)$ which satisfy the conditions
$\alpha-\beta+1>0$, $\alpha\leq 2\beta$ and $\alpha(\beta+1)\leq\beta^2$ except
in the case $\alpha=\beta=0$ which corresponds to the classical isoperimetric
inequality.
