Inequalities from Complex Analysis

✍ Scribed by John P. D'Angelo

• Publisher: The Mathematical Association of America
• Year: 2002
• Tongue: English
• Leaves: 284
• Series: Carus Mathematical Monographs 28
• Category: Library

✦ Synopsis

Inequalities from Complex Analysis is a careful, friendly exposition of some rather interesting mathematics. The author begins by defining the complex number field; he gives a novel presentation of some standard mathematical analysis in the early chapters. The development culminates with some results from recent research literature. The book provides complete yet comprehensible proofs as well as some surprising consequences of the results. One unifying theme is a complex variables analogue of Hilbert's seventeenth problem. Numerous examples, exercises and discussions of geometric reasoning aid the reader. The book is accessible to undergraduate mathematicians, as well as physicists and engineers.

✦ Table of Contents

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
Back Cover