An area theorem for holomorphic functions

Abstract:

Let $D \subsetneqq \mathbf {C}$ and $\Delta \subsetneqq \mathbf {C}$ be open sets in the plane $\mathbf {C}$ containing 0. Suppose that $D$ is connected and $f$ is holomorphic in $D$ with $f(0) = 0$. Then, for $Q = f(D) \cap \Delta$ and $P = {f^{ - 1}}(Q)$, we have \[ A(P) + A(Q) \geqslant \pi {[\min \{ \operatorname {dis}(\partial D,0),\operatorname {dis}(\partial \Delta ,0)\} ]^2},\] $A( \cdot )$ denoting the area. The constant $\pi$ is sharp and the equality holds if and only if $f \equiv 0,\operatorname {dis}(\partial D,0) \leqslant \operatorname {dis}(\partial \Delta ,0)$, and $D$ is a disk of center 0.