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A Schottky-type theorem for starlike domains in Banach spaces. (English) Zbl 1124.46024
This paper in infinite-dimensional holomorphy gives some extensions of the classical Schottky theorem that states, loosely speaking, that the set of all complex-valued holomorphic functions that omit, say, \(0\) and \(1\), is a normal family in the sense of Montel over a disc or a domain. The authors draw upon the classical Schottky theorem and one of its infinite-dimensional generalizations to prove five statements in this paper, of which we state Theorems 1 and 4.
Theorem 1. Let \(U\) be a starlike open subset of a Banach space and for \(n\geq1\) denote by \(U_n\) the set of all \(x\in U\) such that \(\| x\| <n\) and \(U\) contains the ball \(\| y-x\| <1/n\) centered at \(x\). Then there is a function \(c_n(\alpha)>0\), \(\alpha>0\), \(n\geq1\), such that if \(f:U\to{\mathbb C}\setminus\{0,1\}\) is holomorphic and \(| f(0)| \leq\alpha\), then \(| f(x)| <c_n(\alpha)\) for all \(x\in U_n\).
The proof of Theorem 1 considers a starlike domain \(V_n\) with \(U_n\subset V_n\subset U\), various balls, and applies the Schottky-type theorem of P. Takatsuka [Port. Math. (N. S.) 63, No. 3, 351–362 (2006; Zbl 1124.46025); see the following review].
Theorem 4. Let \(U\) be a domain in a Banach space and fix \(x_0\in U\). Then there is a function \(c(a,r,\alpha)>0\), where \(a\in U\), \(\alpha>0\), and \(r>0\) is such that \(U\) contains the ball \(\| x-a\| <r\), with the property that if \(f:U\to{\mathbb C}\setminus\{0,1\}\) is holomorphic and \(| f(x_0)| \leq\alpha\), then \(| f(x)| \leq c(a,r,\alpha)\) for all \(\| x-a\| <r\).
The proof of Theorem 4 looks at a finite chain of balls from \(x_0\) to \(a\) and applies the same Schottky-type result as in Theorem 1.
The paper is well written, and easy to read.

MSC:
46G20 Infinite-dimensional holomorphy
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
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