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An improved estimate for the Bloch norm of functions in Doob's class


Authors: J. S. Hwang and D. C. Rung
Journal: Proc. Amer. Math. Soc. 80 (1980), 406-410
MSC: Primary 30D50
DOI: https://doi.org/10.1090/S0002-9939-1980-0580994-1
MathSciNet review: 580994
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Abstract: For any fixed $ 0 < \rho < 2\pi ,\mathcal{D}(\rho )$ is the family of all holomorphic functions in $ \Delta $ which satisfy (i) $ f(0) = 0$, and (ii) $ {\underline {\lim } _{z \to \tau }}\vert f(z)\vert \geqslant 1$, for all $ \tau $ lying on some arc $ {\Gamma _f} \subseteq \partial \Delta $ with arclength $ \vert{\Gamma _f}\vert \geqslant \rho $. We showed that for each $ f \in \mathcal{D}(\rho )$ there exists a point $ {z_f} \in \Delta $ at which

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

In this paper we improve this estimate by replacing the quantity $ \pi - (\rho /2)$ with a value $ \theta (\rho )$ which lies between 0 and $ \pi - (\rho /2)$ and so improves the estimate. The value $ \theta (\rho )$ is defined as the (unique) solution in this interval of the equation $ {F_\rho }(\theta ) = \log (\cot (\rho /4)\cot (\theta /2)) - \theta /\sin \theta = 0.$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0580994-1
Keywords: Bloch, Landau constant, differential maximal principle
Article copyright: © Copyright 1980 American Mathematical Society

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