Abstract
This chapter is fairly tightly unified around the following simple result:
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Theorem 6.1
(Schwarz’ Lemma) Let f be holomorphic and bounded by 1 in D = D(0, 1) and f(0) = 0. Then
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(i)
∣f′(0)∣ ≤ 1
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(ii)
∣f(z)∣ ≤ ∣z∣ ∀z ∈ D and equality in (i) or in (ii) for some non-zero z occurs if and only if
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(iii)
f(z) = cz for some unimodular complex constant c.
An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-0348-9374-9_16
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© 1979 Birkhäuser Verlag Basel
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Burckel, R.B. (1979). Schwarz’ Lemma and its Many Applications. In: An Introduction to Classical Complex Analysis. Mathematische Reihe, vol 64. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9374-9_7
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DOI: https://doi.org/10.1007/978-3-0348-9374-9_7
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