Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Cluster values of bounded analytic functions
HTML articles powered by AMS MathViewer

by T. W. Gamelin PDF
Trans. Amer. Math. Soc. 225 (1977), 295-306 Request permission

Abstract:

Let D be a bounded domain in the complex plane, and let $\zeta$ belong to the topological boundary $\partial D$ of D. We prove two theorems concerning the cluster set ${\text {Cl}}(f,\zeta )$ of a bounded analytic function f on D. The first theorem asserts that values in ${\text {Cl}}(f,\zeta )\backslash f(\Sha _\zeta )$ are assumed infinitely often in every neighborhood of $\zeta$, with the exception of those lying in a set of zero analytic capacity. The second asserts that all values in ${\text {Cl}}(f,\zeta )\backslash f({\mathfrak {M}_\zeta } \cap {\text {supp}}\;\lambda )$ are assumed infinitely often in every neighborhood of $\zeta$, with the exception of those lying in a set of zero logarithmic capacity. Here ${\mathfrak {M}_\zeta }$ is the fiber of the maximal ideal space $\mathfrak {M}(D)$ of ${H^\infty }(D)$ lying over $\zeta$, $\Sha _\zeta$ is the Shilov boundary of the fiber algebra, and $\lambda$ is the harmonic measure on $\mathfrak {M}(D)$.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 46J15, 30A72
  • Retrieve articles in all journals with MSC: 46J15, 30A72
Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 225 (1977), 295-306
  • MSC: Primary 46J15; Secondary 30A72
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0438133-8
  • MathSciNet review: 0438133