Some isoperimetric inequalities in the plane with radial power weights

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 $alpha-beta+1>0$, $alphaleq 2beta$ and $alpha(beta+1)leqbeta^2$ except in the case $alpha=beta=0$ which corresponds to the classical isoperimetric inequality.