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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the ranges of analytic functions
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by J. S. Hwang PDF
Trans. Amer. Math. Soc. 260 (1980), 623-629 Request permission

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 $\left | A \right | \geqslant 2\rho > 0$, \[ \lim \inf \limits _{i \to \infty } \left | {f({P_i})} \right | \geqslant 1 {\text {where}} {P_i} \to P \in A {\text {and}} {P_i} \in U.\] 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 \[ (1 - \left | z \right |)\left | {{f_n}’ (z)} \right | \leqslant 1/n {\text {for}} z \in U, {\text {where}} n > N(\rho ).\] The function \[ {f_n}(z) = 1 + (1 - {z^n})/{n^2},\] 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 \[ N(r, \rho ) \doteqdot (1/(1 - r))\log (1/(1 - \cos \rho )).\] .
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 260 (1980), 623-629
  • MSC: Primary 30D40
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0574804-0
  • MathSciNet review: 574804