| Contents | 7 |
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| Introduction | 9 |
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| 1.1 Historical remarks | 9 |
| 1.2 On inequalities for higher derivatives | 11 |
| 1.3 On methods | 13 |
| 1.4 Survey of the contents | 14 |
| Basic coefficient inequalities | 15 |
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| 2.1 Subordinate functions | 15 |
| 2.2 Bieberbach s conjecture by de Branges | 19 |
| 2.3 Theorems of Jenkins and Sheil-Small | 23 |
| 2.4 Inverse coefficients | 26 |
| 2.5 Domains with bounded boundary rotation | 31 |
| The Poincar´e metric | 35 |
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| 3.1 Background | 35 |
| 3.2 The Schwarz-Pick inequality | 38 |
| 3.3 Estimates using the Euclidean distance | 41 |
| 3.4 An application of Teichmüller s theorem | 45 |
| 3.5 Domains with uniformly perfect boundary | 48 |
| 3.6 Derivatives of the conformal radius | 52 |
| Basic Schwarz-Pick type inequalities | 57 |
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| 4.1 Two classical inequalities | 58 |
| 4.2 Theorems of Ruscheweyh and Yamashita | 60 |
| 4.3 Pairs of simply connected domains | 63 |
| 4.4 Holomorphic mappings into convex domains | 67 |
| 4.5 Punishing factors for convex pairs | 71 |
| 4.6 Case n = 2 for all domains | 74 |
| Punishing factors for special cases | 77 |
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| 5.1 Solution of the Chua conjecture | 77 |
| 5.2 Punishing factors for angles | 80 |
| 5.3 Sharp lower bounds for punishing factors | 86 |
| 5.4 Domains in the extended complex plane | 92 |
| 5.5 Maps from convex into concave domains | 98 |
| Multiply connected domains | 104 |
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| 6.1 Finitely connected domains | 104 |
| 6.2 Pairs of arbitrary domains | 111 |
| 6.3 Some examples | 115 |
| Related results | 120 |
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| 7.1 Inequalities for schlicht functions | 120 |
| 7.2 Derivatives of a-invariant functions | 124 |
| 7.3 A characterization of convex domains | 131 |
| Some open problems | 134 |
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| 8.1 The Krzyz conjecture | 134 |
| 8.2 The angle conjecture | 135 |
| 8.3 The generalized Goodman conjecture | 138 |
| 8.4 Bloch and several variable problems | 146 |
| 8.5 On sums of inverse coefficients | 147 |
| Bibliography | 150 |
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| Index | 161 |
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