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Starlike and convex rational mappings on infinite dimensional domains. (English) Zbl 1163.32003

Let \(\Delta=\left\{\zeta\in\mathbb{C}:|\zeta|<1\right\}\) be the open unit disc in \(\mathbb{C}\). Let \((E,\left\|\cdot\right\|)\) be a complex Banach space with dual \(E^{*}\). The open unit ball \(\left\{x\in E:\left\|x\right\|<1\right\}\) is denoted by \(B\).
Consider a holomorphic function \(\chi:\Delta\rightarrow\mathbb{C}\setminus\left\{0\right\}\) of the form \[ \chi(\zeta)=1+\sum_{k=1}^{\infty}a_{k}\zeta^{k}\;\;(a_{k}\in\mathbb{C}). \]
Let \(f:B\rightarrow E\) be a holomorphic mapping given by
\[ f(z)=\frac{z}{g(z)}\;\;\text{and}\;\;g(z)=\chi\circ\psi(z),\tag{1} \]
where \(\psi\in E^{*}\) with \(\left\|\psi\right\|=1\).
In the paper, the authors give conditions on the coefficients \(a_{k}\) for \(f\) to be starlike, strongly starlike of order \(\alpha\;(0<\alpha\leq1)\) and starlike of order \(\gamma\;(0<\gamma<1)\). They also give a sufficient condition for mappings \(f\) of the form (1) to be convex when they are defined in Hilbert spaces.

MSC:

32A30 Other generalizations of function theory of one complex variable
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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