  We prove, using elementary methods of complex analysis, the following
generalization of the isoperimetric inequality: if $p\in\re$,
$\Omega\subset\re^2$ then the inequality $$
  \left(\frac{|\Omega|}{\pi}\right)^{\frac{p+1}{2}}\leq\frac{1}{2\pi}\int_{\delomega}|x|^pd\sigma(x)
$$ holds true under appropriate assumptions on $\Omega$ and $p.$ This solves an
open problem arising in the context of isoperimetric problems with density and
poses some new ones (for instance generalizations to $\re^n$). We prove the
equivalence with a Hardy-Sobolev inequality, giving the best constant, and
generalize thereby the equivalence between the classical isoperimetric
inequality and the Sobolev inequality. Furthermore, the inequality paves the
way for solving another problem: the generalization of the harmonic
transplantation method of Flucher to the singular Moser-Trudinger embedding.
