The areas of polynomial images and pre-images

Let p be a monic complex polynomial of degree n, and let K be a measurable subset of the complex plane. Then the area of p(K), counted with multiplicity, is at least pi.n(Area(K)/pi)^n, and the area of the pre-image of K under p is at most (pi^(1-1/n)).(Area(K))^(1/n). Both bounds are sharp. The special case of the pre-image result in which K is a disc is a classical result, due to Polya. The proof is based on Carleman's classical isoperimetric inequality for plane condensers.