Distortion of sets by inner functions

Abstract:

For any inner function $f$ with $f(0) = 0$ and any Borel set $E \subset {\mathbf {D}}$ \[ {M_\alpha }(z \in {\mathbf {D}}:f(z) \in E) \geqslant {M_\alpha }(E),\qquad 0 < \alpha \leqslant 1,\] where ${M_\alpha }$ denotes $\alpha$-dimensional Hausdorff measure. In the case that $0 < {M_\alpha }(E) < \infty$ we have equality only for rotations of the identity.