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 strongly convex 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