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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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##### References:
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