## Some Extremal Problems in Conformal and Quasiconformal Mapping |

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### Contents

GRUNSKYS POTENTIAL THEORETIC INEQUALITY | 7 |

THE EXTREMAL PROBLEM FOR CONFORMAL MAPPINGS | 13 |

THE EXTREMAL PROBLEM FOR QUASI CONFORMAL MAPPINGS | 28 |

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analytic asymptotic expansion bound boundary components calculate X[f Cauchy-Riemann equations 3.7 Chapter compact subsets complement consists conformal mapping function constant Corollary defined degenerate div(K grad Doctor of Philosophy domain dependent quantities equals essential supremum extended complex plane extremal problem f uniformly foci for f function f function which maps G M-N GM-N Green's function Green's identity Grunsky Grunsky's theorem harmonic conjugate harmonic function harmonic measures interior radius j e M-N J K jGN j,k=l J J j,kGM j,kGM-N j,kGN jGM-N Jk j,k=l K|grad lemniscate given log|f(z log|w-w log|z log|z-z maximum principle n-tuply connected domain nondegenerate function obtain optimal foci parameters period matrix potential theoretic inequality proof of Lemma proof of Theorem prove quasiconformal mapping function real numbers restrictions on K(z satisfies 3.2 Schiffer and Schober Schober 18 simply connected smooth uniformly on compact variational methods Xj log zero