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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.

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Bounded holomorphic functions on bounded symmetric domains
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by Joel M. Cohen and Flavia Colonna PDF
Trans. Amer. Math. Soc. 343 (1994), 135-156 Request permission


Let D be a bounded homogeneous domain in ${\mathbb {C}^n}$, and let $\Delta$ denote the open unit disk. If $z \in D$ and $f:D \to \Delta$ is holomorphic, then ${\beta _f}(z)$ is defined as the maximum ratio $|{\nabla _z}(f)x|/{H_z}{(x,\bar x)^{1/2}}$, where x is a nonzero vector in ${\mathbb {C}^n}$ and ${H_z}$ is the Bergman metric on D. The number ${\beta _f}(z)$ represents the maximum dilation of f at z. The set consisting of all ${\beta _f}(z)$ for $z \in D$ and $f:D \to \Delta$ holomorphic, is known to be bounded. We let ${c_D}$, be its least upper bound. In this work we calculate ${c_D}$ for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions f for which ${\beta _f} = {c_D}$) when D contains $\Delta$ as a factor, and show that the class of extremal functions is very large when $\Delta$ is not a factor of D.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 343 (1994), 135-156
  • MSC: Primary 32A37; Secondary 32M15, 46E15
  • DOI:
  • MathSciNet review: 1176085