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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Boundary zero sets of $A^\infty$ functions satisfying growth conditions
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by B. A. Taylor and D. L. Williams PDF
Proc. Amer. Math. Soc. 35 (1972), 155-160 Request permission

Abstract:

Let A denote the algebra of functions analytic in the open unit disc D and continuous in D, and let \[ {A^\infty } = \{ f \in A:{f^{(n)}} \in A,n = 0,1,2, \cdots \} .\] For $f \in A$ denote the set of zeros of f in D by ${Z^0}(f)$, and for $f \in {A^\infty }$ let ${Z^\infty }(f) = \bigcap {_{n = 0}^\infty Z{^0}({f^{(n)}})}$. We study the boundary zero sets of ${A^\infty }$ functions F satisfying, for some sequence $\{ {M_n}\}$ and some $B > 0$, \begin{equation}\tag {$1$}|{F^{(n)}}(z)| \leqq n!{B^n}{M_n},\quad z \in \bar D,n = 0,1,2, \cdots .\end{equation} In particular, when ${M_n} = \exp ({n^p}),p > 1$, it is shown that for E, a proper closed subset of $\partial D$, there exists $F \in {A^\infty }$ satisfying (1) and with ${Z^0}(F) = {Z^\infty }(F) = E$ if and only if $\smallint _{ - \pi }^\pi |\log \rho ({e^{i\theta }},E){|^q}D\theta < + \infty$. Here $\rho (z,E)$ is the distance from z to E and $(1/p) + (1/q) = 1$.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 35 (1972), 155-160
  • MSC: Primary 30A72
  • DOI: https://doi.org/10.1090/S0002-9939-1972-0310253-9
  • MathSciNet review: 0310253