Topics in Hardy Classes and Univalent Functions

1 Harmonic Functions.- 1.1 Introduction.- 1.2 Uniqueness principle.- 1.3 The Poisson kernel.- 1.4 Normalized Lebesgue measure.- 1.5 Dirichlet problem for the unit disk.- 1.6 Properties of harmonic functions.- 1.7 Mean value property.- 1.8 Harnack's theorem.- 1.9 Weak compactness principle.- 1.10 Nonnegative harmonic functions.- 1.11 Herglotz and Riesz representation theorem.- 1.12 Stieltjes inversion formula.- 1.13 Integral of the Poisson kernel.- 1.14 Examples.- 1.15 Space h1(D).- 1.16 Characterization of h1(D).- 1.17 Nontangential convergence.- 1.18 Fatou's theorem.- 1.19 Boundary functions.- Examples and addenda.- 2 Subharmonic Functions.- 2.1 Introduction.- 2.2 Upper semicontinuous functions.- 2.3 Subharmonic functions.- 2.4 Some properties of subharmonic functions.- 2.5 Maximum principle.- 2.6 Convergence of mean values.- 2.7 Convex functions.- 2.8 Structure of convex functions.- 2.9 Jensen's inequality.- 2.10 Composition of convex and subharmonic functions.- 2.11 Vector- and operator-valued functions.- 2.12 Subharmonic functions from holomorphic functions.- 3 Part I Harmonic Majorants Part II Nevanlinna and Hardy-Orlicz Classes.- 3.1 Introduction.- 3.2 Least harmonic majorant.- 3.3 Existence of least harmonic majorants.- 3.4 Construction of harmonic majorants.- 3.5 Class shl(D).- 3.6 Characterization of sh1(D).- 3.7 Absolutely continuous component of a related measure.- 3.8 Uniformly integrable family.- 3.9 Strongly convex functions.- 3.10 Theorem of de la Vallée Poussin and Nagumo.- 3.11 Singular component of associated measures.- 3.12 Sufficient conditions for absolute continuity.- 3.13 Theorem of Szegö-Solomentsev.- 3.14 Remark.- 3.15 Hardy and Nevanlinna classes.- 3.16 Linearity of the classes.- 3.17 Properties of log+x.- 3.18 Majorants for stronglyconvex functions.- 3.19 Compositions and restrictions.- 3.20 Quotients of bounded functions.- Examples and addenda.- 4 Hardy Spaces on the Disk.- 4.1 Introduction.- 4.2 Inner and outer functions.- 4.3 Rational inner functions.- 4.4 Infinite products.- 4.5 An infinite product.- 4.6 Blaschke products.- 4.7 Inner functions with no zeros.- 4.8 Singular inner functions.- 4.9 Factorization of inner functions.- 4.10 Boundary functions for N(D).- 4.11 Characterization of N(D).- 4.12 Condition on zeros.- 4.13 N(D) as an algebra.- 4.14 Characterization of N+(D).- 4.15 N+(D) as an algebra.- 4.16 Estimates from boundary functions for N+(D).- 4.17 Outer functions in N+(D).- 4.18 Characterization of ??(D).- 4.19 Nevanlinna and Hardy-Orlicz classes on the boundary.- 4.20 Szegö's problem.- 4.21 Classes HP(D) and HP(?).- 4.22 Characterization of HP(D).- 4.23 Characterization of HP(?).- 4.24 Connection between HP(D) and HP(?).- 4.25 Hp(?) as a subspace of LP(?).- 4.26 Hp(D) and HP(?) as Banach spaces.- 4.27 F and M Riesz theorem.- 4.28 H2(D) and H2(?).- 4.29 Sufficient conditions for outer functions.- 4.30 Beurling's theorem.- 4.31 Theorem of Szegö, Kolmogorov, and Kre?n.- 4.32 Closure of trigonometric functions in Lp(?).- 5 Function Theory on a Half-Plane.- 5.1 Introduction.- 5.2 Poisson representation.- 5.3 Nevanlinna representation.- 5.4 Stieltjes inversion formula.- 5.5 Fatou's theorem.- 5.6 Boundary functions for N(?).- 5.7 Limits of nondecreasing functions.- 5.8 Nonnegative harmonic functions.- 5.9 Theorem of Flett and Kuran.- 5.10 Nevanlinna and Hardy-Orlicz classes.- 5.11 Notation and terminology.- 5.12 Szegö's problem on the line.- 5.13 Inner and outer functions.- 5.14 Examples and miscellaneous properties.- 5.15 Hardy classes.- 5.16 Characterization of?P(I?).- 5.17 Inclusions among classes.- 5.18 Poisson representation for ?P(?).- 5.19 Cauchy representation for Hp(?).- 5.20 Characterization of HP(?).- 5.21 Hp(?) as a subspace of N+(?).- 5.22 Condition for mean convergence.- 5.23 Hp(?)and ?P(?) as subspaces of N+(?).- 5.24 HP(?)and ?p(?) as Banach spaces.- 5.25 Local convergence to a boundary function.- 5.26 Remark on the definition of HP(?).- 5.27 Plancherel theorem.- 5.28 Paley-Wiener representation.- 5.29 Natural isomorphisms.- 5.30 Hilbert transforms.- 5.31 Real and imaginary parts of boundary functions.- 5.32 Cauchy transform on Lp(??, ?).- 5.33 Mapping f? f-i f on Lp(-?, ?) to HP(R).- 5.34 M Riesz theorem.- 5.35 Algebraic properties of Hilbert transforms.- Examples and addenda.- 6 Phragmén-Lindelöf Principle.- 6.1 Introduction.- 6.2 Phragmén-Lindelöf principle.- 6.3 Functions on a sector.- 6.4 Estimate from behavior on the imaginary axis.- 6.5 Blaschke products on the imaginary axis.- 6.6 Equivalence of the unit disk and a half-disk.- 6.7 Function theory on a half-disk.- 6.8 Estimates on a half-disk.- 6.9 Test to belong to N(?).- 6.10 Asymptotic behavior of Poisson integrals.- 6.11 Estimate from behavior on semicircles.- 6.12 Blaschke products on semicircles.- 6.13 Factorization of bounded type functions.- 6.14 Nevanlinna factorization and mean type.- 6.15 Formulas for mean type.- 6.16 Exponential type.- 6.17 Kre?n's theorem.- 6.18 Inequalities for mean type.- Examples and addenda.- 7 Loewner Families.- 7.1 Definitions and overview of the subject.- 7.2 Preliminary results.- 7.3 Riemann mapping theorem.- 7.4 The Dirichlet space and area theorem.- 7.5 Generalization of the Dirichlet space.- 7.6 Bieberbach's theorem.- 7.7 Size of the image domain.- 7.8Distortion theorem.- 7.9 Carathéodory convergence theorem.- 7.10 Subordination.- 7.11 Technical lemmas.- 7.12 Parametric representation of Loewner families.- 8 Loewner's Differential Equation.- 8.1 Loewner families and associated semigroups.- 8.2 Estimates derived from Schwarz's lemma.- 8.3 Absolute continuity.- 8.4 Herglotz functions.- 8.5 Loewner's differential equation.- 8.6 Solution of the nonlinear equation.- 8.7 Solution of Loewner's differential equation.- 9 Coefficient Inequalities.- 9.1 Three famous problems.- 9.2 de Branges' method.- 9.3 Construction of the weight functions.- 9.4 Askey-Gasper inequality.- Notes.