Abstract
In the paper [B2], Baernstein constructs a simply connected domain $\Omega$ in the plane for which the conformal mapping $f$ of $\Omega$ into the unit disc $\Delta$ satisfies $$\int_{\mathbf{R} \cap \Omega} |f'(z)|^p |dz| = \infty,$$ for some $p ∈ (1,2)$, where $\mathbf R$ is the real line.
This gives a counterexample to a conjecture stating that for any simply connected domain $\Omega$ in the plane, all the above integrals are finite for any $1 \lt p \lt 2$.
In this paper, we give a conceptual proof of the basic estimate of Baenstein.
Acknowledgment
I would like to thank Professor Albert Baernstein II for his helpful comments and suggestions concerning this work.
Citation
Enrique Villamor. "On a theorem of Baernstein." Ark. Mat. 33 (1) 183 - 197, 1995. https://doi.org/10.1007/BF02559610
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