Boundary differential relations for holomorphic functions on the disc

Černe, Miran; Zajec, Matej

doi:10.1090/S0002-9939-2010-10469-0

Boundary differential relations for holomorphic functions on the disc
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by Miran Černe and Matej Zajec PDF

Proc. Amer. Math. Soc. 139 (2011), 473-484 Request permission

Abstract:

The existence of solutions of boundary differential relations for holomorphic functions on the disc $\Delta$ is considered. First we prove that for an arbitrary continuous positive function $\Phi$ on the complex plane $\mathbb {C}$ there exists a disc algebra function $f\in A(\Delta )$ such that $|f^{\prime }|=\Phi (f)$ on $\partial \Delta$. Assuming some smoothness, the existence result is also proved for a quite general differential relation $\rho (\xi ,f^{\prime }(\xi ))=\Phi (\xi , f(\xi ))$, $\xi \in \partial \Delta$, where $\rho$ is a defining function for a family of Jordan curves in $\mathbb {C}$ containing point $0$ in its interior and $\Phi$ is a bounded positive function on $\partial \Delta \times \mathbb {C}$.

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