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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Starlike, convex, close-to-convex, spiral-like, and $ \Phi $-like maps in a commutative Banach algebra with identity


Authors: L. F. Heath and T. J. Suffridge
Journal: Trans. Amer. Math. Soc. 250 (1979), 195-212
MSC: Primary 46J15; Secondary 30C45, 46G99
MathSciNet review: 530050
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Abstract: Let C(X) be the space of continuous functions on a compact $ {T_2}$-space X where each point of X is a $ {G_\delta }$. If $ F:\,B\, \to \,C\,(X)$ is a biholomorphic (in the sense that F and $ {F^{ - 1}}$ are Fréchet differentiable) map of $ B\, = \,\{ \,f\left\vert {\,\left\Vert f \right\Vert} \right.\, < \,1\} $ onto a convex domain with $ DF(0)\, = \,I$, then F is Lorch analytic (i.e., $ DF\,(f)(g)\, = \,{a_f}g $ for some $ {a_{f}} \, \in \,C\,(X))$). Let R be a commutative Banach algebra with identity such that the Gelfand homomorphism of R into $ C(\mathcal{m})$ is an isometry. Starlike, convex, close-to-convex, spirallike and $ \Phi $-like functions are defined in $ B\, = \,\{ x\, \in \,R\,\left\vert {\,\left\Vert x \right\Vert} \right.\, < \,1\} $ for L-analytic functions in B and they are related to associated complex-valued holomorphic functions in $ \Delta \, = \,\{ z\, \in \,\left. {\textbf{C}} \right\vert\,\,\left\vert z \right\vert\, < \,1\} $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0530050-X
PII: S 0002-9947(1979)0530050-X
Keywords: Starlike, convex, close-to-convex, spirallike, $ \Phi $-like, F-holomorphic, L-analytic
Article copyright: © Copyright 1979 American Mathematical Society