  Let vphi:C rightarrow C be a bilipschitz map. We prove that if E\subset\C is
compact, and gamma(E), alpha(E) stand for its analytic and continuous analytic
capacity respectively, then C^{-1}\gamma(E)\leq \gamma(\vphi(E)) \leq
C\gamma(E) and C^{-1}\alpha(E)\leq \alpha(\vphi(E)) \leq C\alpha(E), where C
depends only on the bilipschitz constant of vphi. Further, we show that if mu
is a Radon measure on C and the Cauchy transform is bounded on L^2(\mu), then
the Cauchy transform is also bounded on L^2(\vphi_\sharp\mu), where
vphi_\sharp\mu is the image measure of mu by vphi. To obtain these results, we
estimate the curvature of vphi_\sharp\mu by means of a corona type
decomposition.
