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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


On the ranges of analytic functions

Author: J. S. Hwang
Journal: Trans. Amer. Math. Soc. 260 (1980), 623-629
MSC: Primary 30D40
MathSciNet review: 574804
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Abstract: Following Doob, we say that a function $ f(z)$ analytic in the unit disk U has the property $ K(\rho )\,$ if $ f(0)\, = \,0$ and for some $ \operatorname{arc} \,A\,$ on the unit circle whose measure $ \left\vert A \right\vert\, \geqslant \,2\rho \, > \,0$,

$\displaystyle \mathop {\lim \,\inf }\limits_{i \to \infty } \,\left\vert {f({P_... ... \,1\,{\text{where}}\,{P_i}\, \to \,P\, \in \,A\,{\text{and}}\,{P_i}\, \in \,U.$

We recently have solved a problem of Doob by showing that there is an integer $ N(\rho )$ such that no function with the property $ K(\rho )$ can satisfy

$\displaystyle (1\, - \,\left\vert z \right\vert)\left\vert {{f_n}' (z)} \right\... ... \leqslant \,1/n\,{\text{for}}\,z\, \in \,U,\,{\text{where}}\,n\, > \,N(\rho ).$

The function

$\displaystyle {f_n}(z)\, = \,1\, + \,(1\, - \,{z^n})/{n^2},$

shows that the condition $ {f_n}(0)\, = \,0$ is necessary and cannot be replaced by $ {f_n}(0)\, = \,r{e^{i\alpha }}$, for $ r\, > \,1$. Naturally, we may ask whether this can be replaced by $ {f_n}(0)\, = \,r{e^{i\alpha }}$, for $ r\, < \,1$? The answer turns out to be yes, when $ n\, > \,N\,(r,\,\rho )$, where

$\displaystyle N(r,\,\rho )\,\doteqdot\,(1/(1\, - \,r))\log (1/(1\, - \,\cos \rho )).$


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Additional Information

PII: S 0002-9947(1980)0574804-0
Keywords: The range and analytic function
Article copyright: © Copyright 1980 American Mathematical Society

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