## Boundary zero sets of $A^\infty$ functions satisfying growth conditions

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- by B. A. Taylor and D. L. Williams
- Proc. Amer. Math. Soc.
**35**(1972), 155-160 - DOI: https://doi.org/10.1090/S0002-9939-1972-0310253-9
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## 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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## Bibliographic 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