Inequalities from Complex Analysis

Front Matter
     Cover
     The Carus Mathematical Monographs
     Copyright
          © 2002 by The Mathematical Association of America
          Complete Set ISBN 0-88385.000-1
          Vol. 28 ISBN 0-88385-033.8
          LCCN 2002101375
     Inequalities from Complex Analysis
     Editorial Board
     List of Publishe Monographs
     Contents
     Preface

CHAPTER I  Complex Numbers
     I.1 The real number system
     I.2 Definition of the complex number field
     I.3 Elementary complex geometry
     I.4 Alternative definitions of the complex numbers
          I.4.1 Using matrices
          I.4.2 Using polynomials
     I.5 Completeness
     I.6 Convergence for power series
     I.7 Trigonometry
     I.8 Roots of unity
     I.9 Summary

CHAPTER II  Complex Euclidean Spaces and Hilbert Spaces
     II.1 Hermitian inner products
     II.2 Orthogonality, projections and closed subspaces
     II.3 Orthonormal expansion
     II.4 The polarization identity
     II.5 Generating functions and orthonormal systems

CHAPTER III  Complex Analysis in Several Variables
     III.1 Holomorphic functions
     III.2 Some calculus
     III.3 The Bergman kernel function

CHAPTER IV Linear Transformations and Positivity Conditions
     IV.1 Adjoints and Hermitian forms
     IV.2 Solving linear equations
     IV.3 Linearization
     IV.4 Eigenvalues and the spectral theorem in finite dimensions
     IV.5 Positive definite linear transformations in finite dimensions
     IV.6 Hilbert\'s inequality
     IV.7 Additional inequalities from Fourier analysis

CHAPTER V Compact and Integral Operators
     V:1 Convergence properties for bounded linear transformations
     V.2 Compact operators on Hilbert space
     V.3 The spectral theorem for compact Hermitian operators
     V.4 Integral operators
     V.5 A glimpse at singular integral operators

CHAPTER VI  Positivity Conditions for Real-valued Functions
     VI.1 Real variables analogues
     VI.2 Real-valued polynomials on C^n
     VI.3 Squared norms and quotients of squared norms
     VI.4 Plurisubharmonic functions
     VI.5 Positivity conditions for polynomials

CHAPTER VII  Stabilization and Applications
     VII.1 Stabilization for positive bihomogeneous polynomials
     VII.2 Positivity everywhere
     VII.3 Positivity on the unit sphere
     VII.4 Applications to proper holomorphic mappings between balls
     VII.5 Positivity on zero sets
     VlI.6 Proof of stabilization

CHAPTER VIII Afterword

Back Matter
     APPENDIX A
          A.1 Algebra
          A.2 Analysis
     Bibliography
     Index
     Author
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