The Size of an Analytic Function as Measured by Levy's Time Change

Abstract

For $f$ analytic in the unit disc put $\nu(f) = \int^\tau_0 |f'(B(s))|^2 ds$ where $\tau$ is the exit time of Brownian motion $B(t)$ from the disc. We prove that $E\Phi(\tau) \leq E\Phi(\nu(f))$ for all $f$ satisfying $|f'(0)| = 1$ and a wide class of $\Phi$. In particular, we may take $\Phi(\lambda) = |\lambda|^P$ for $0 < p < \infty$.