Abstract
The center-piece of this chapter is the following remarkable result due to Friedrich Schottky: if f is holomorphic in D(0, 1) and assumes neither of the values 0 or 1, then f is bounded in D(0, r) by a bound depending on r and ∣f(0)∣ only, for each r < 1. [In fact we prove a generalization in which f does not assume 0 and f(n) (for some n > 0) does not assume 1.] The principal application is the almost immediate fact that a family of holomorphic functions on a common domain, none of which has 0 or 1 in its range, if bounded at a point, is uniformly bounded in a neighborhood of that point. The compactness theorems of Chapter VII then come into play with astounding consequences. Of course “0” and “1” here are convenient normalizations: any two distinct complex numbers would serve as well.
An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-0348-9374-9_22
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© 1979 Birkhäuser Verlag Basel
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Burckel, R.B. (1979). Omitted Values and Normal Families. In: An Introduction to Classical Complex Analysis. Mathematische Reihe, vol 64. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9374-9_13
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DOI: https://doi.org/10.1007/978-3-0348-9374-9_13
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9376-3
Online ISBN: 978-3-0348-9374-9
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