  Quasiconformal maps in the complex plane are homeomorphisms that satisfy
certain geometric distortion inequalities; infinitesimally, they map circles to
ellipses with bounded eccentricity. The local distortion properties of these
maps give rise to a certain degree of global regularity and H\"older
continuity. In this paper, we give improved lower bounds for the H\"older
continuity of these maps; the analysis is based on combining the isoperimetric
inequality with a study of the length of quasicircles. Furthermore, the
extremizers for H\"older continuity are characterized, and some applications
are given to solutions to elliptic partial differential equations.
