## Local holomorphic families of discs |

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### Contents

A theorem of Ahlfors | 14 |

2 Ahlfors theorem | 31 |

Proof of Theorem 1 1 | 68 |

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### Common terms and phrases

1+ax 1+ay Ahlfors Banach algebra Banach space belongs Bers fiber space calculation cf^a chain rule change variables Chapter complex anti-linear complex linear composition lemma cp(x cp(z Cq+1 CS(X D2cp define denote depends derivative estimates Df(x diffeomorphism cp differentiable direct sum direct sum decomposition equation Euclidean sense exp(f fact family of discs fiber over Xq formula fractions lemma Holder condition holomorphic family holomorphic function holomorphic map Hom(X,Y homeomorphism homotopy identity integrability condition invertible isometry isomorphism linear transformation maximum modulus principle measurable functions Mobius transformation modified Hilbert transform norm notation open set open subset pointwise proceed by induction product complex structure product lemma Proof Proposition 2.20 pullback Riemann surface spaces of functions Subsection suppose Taylor series theorem 1.1 thesis universal Teichmuller space vector spaces Vekua zero