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Proof of a conjecture of Doob


Authors: J. S. Hwang and D. C. Rung
Journal: Proc. Amer. Math. Soc. 75 (1979), 231-234
MSC: Primary 30D99
DOI: https://doi.org/10.1090/S0002-9939-1979-0532142-3
MathSciNet review: 532142
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Abstract: Let $ \mathcal{D}(\rho )$ be the class of all holomorphic functions f in the unit disc $ \Delta $ such that $ f(0) = 0$ and there exists an arc $ {\Upsilon _f} \subseteq \partial \Delta $ with length $ \vert{\Upsilon _f}\vert \geqslant \rho $ such that $ \underline {\lim } \vert f(z)\vert \geqslant 1,z \to \tau \in {\Upsilon _f}$. In 1935, J. L. Doob asked, in essence, whether the Bloch norms $ \{ \left\Vert f \right\Vert = {\sup _{z \in \Delta }}\vert f'(z)\vert(1 - \vert z{\vert^2})\} $ have a positive lower bound for the class $ \mathcal{D}(\rho )$. We show that if $ f \in \mathcal{D}(\rho )$ there exists a $ {z_f} \in \Delta $ such that

$\displaystyle \vert f'({z_f})\vert\left( {1 - \vert{z_f}{\vert^2}} \right) \geqslant \frac{2}{e}\;\frac{{\sin (\pi - \rho /2)}}{{(\pi - \rho /2)}}.$


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0532142-3
Keywords: Bloch, Landau constants, differential maximal principle
Article copyright: © Copyright 1979 American Mathematical Society

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