Derivatives of Inner Functions

Cover......Page 1
Derivatives of Inner Functions......Page 4
Preface......Page 8
Contents......Page 10
1.1 The Poisson Integral of a Measure......Page 12
1.2 The Hardy Space Hp(D)......Page 17
1.3 Two Classes of Inner Functions......Page 19
1.4 The Canonical Factorization......Page 23
1.5 A Characterization of Blaschke Products......Page 28
1.6 The Nevanlinna Class N and Its Subclass N+......Page 31
1.7 Bergman Spaces......Page 34
2.1 Frostman Shifts and the Exceptional Set ε......Page 38
2.2 Capacity......Page 41
2.3 Hausdorff Dimension......Page 43
2.4 ε Has Logarithmic Capacity Zero......Page 46
2.5 The Cluster Set at a Boundary Point......Page 48
3.1 Elementary Formulas for B'......Page 50
3.2 The Cardinality of the Zeros of B'......Page 51
3.3 A Formula for |B'|......Page 53
3.4 The Locus of the Zeros of B' in D......Page 57
3.5 B Has a Nonzero Residue......Page 59
4.1 Elementary Formulas for B' and S'......Page 62
4.2 Some Estimations for Hp-Means......Page 65
4.3 Some Estimations for Ap-Means......Page 68
4.4 The Angular Derivative......Page 71
4.5 The Carathéodory Derivative......Page 73
4.6 Another Characterization of the Carathéodory Derivative......Page 79
5.1 The Effect of Singular Factors......Page 82
5.2 A Characterization of Φ' Hp(D)......Page 84
5.3 We Never Have S' H12(D)......Page 85
5.4 The Distance Function......Page 87
5.5 A Construction of S with S' Hp(D) for All 06.1 We Always Have Φ' ∩06.2 When Does Φ' Bp(D) for Some 12 p <1?......Page 94
6.3 We Never Have S' B23(D)......Page 97
6.4 A Construction of S with S' Bp(D) for All 06.5 A Generalized Cantor Set......Page 103
6.6 An Example of S with S' Bp(D) for All 07.1 Frostman's Theorem, Local Version......Page 108
7.2 The Radial Variation......Page 111
7.3 Frostman's Theorems, Global Version......Page 116
7.4 An Example of B with B' N......Page 123
7.5 A Sufficient Condition for the Existence of B'(ei)......Page 126
7.6 The Global Behavior of B'......Page 128
8.1 When Do We Have B' H1(D)?......Page 134
8.2 A Sufficient Condition for B' Hp(D), 08.3 What Does B' Hp(D), 08.4 Some Examples of Blaschke Products......Page 143
8.5 The Study of B'p When Zeros Are in a Stolz Domain......Page 147
8.6 The Effect of Argument of Zeros on B'p......Page 151
9.1 A Sufficient Condition for B' Bp(D)......Page 153
9.2 What Does B' Bp(D) Imply?......Page 155
9.3 An Example of Blaschke Products......Page 159
9.4 The Effect of Argument of Zeros on B' Bp......Page 160
10.1 An Estimation Lemma......Page 164
10.2 Hp-Means of the First Derivative......Page 166
10.3 Hp-Means of Higher Derivatives......Page 169
10.4 Ap-Means of the First Derivative......Page 171
References......Page 174
Index......Page 176