Proof of a conjecture of Doob
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- by J. S. Hwang and D. C. Rung PDF
- Proc. Amer. Math. Soc. 75 (1979), 231-234 Request permission
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 $|{\Upsilon _f}| \geqslant \rho$ such that $\underline {\lim } |f(z)| \geqslant 1,z \to \tau \in {\Upsilon _f}$. In 1935, J. L. Doob asked, in essence, whether the Bloch norms $\{ \left \| f \right \| = {\sup _{z \in \Delta }}|f’(z)|(1 - |z{|^2})\}$ 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 \[ |f’({z_f})|\left ( {1 - |{z_f}{|^2}} \right ) \geqslant \frac {2}{e}\;\frac {{\sin (\pi - \rho /2)}}{{(\pi - \rho /2)}}.\]References
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Additional Information
- © Copyright 1979 American Mathematical Society
- 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