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Topics in Hardy Classes and Univalent Functions

These notes are based on lectures given at the University of Virginia over the past twenty years. They may be viewed as a course in function theory for nonspecialists. Chapters 1-6 give the function-theoretic background to Hardy Classes and Operator Theory, Oxford Mathematical Monographs, Oxford Uni...

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Bibliographic Details
Main Authors: Rosenblum, Marvin, Rovnyak, James (Author)
Format: eBook
Language:English
Published: Basel Birkhäuser 1994, 1994
Edition:1st ed. 1994
Series:Birkhäuser Advanced Texts Basler Lehrbücher
Subjects:
Functional Analysis
Functions Of Complex Variables
Mathematical Analysis
Analysis
Functions Of A Complex Variable
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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
  • 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
  • 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(?)
  • 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
  • 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

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