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.

lgrsnf/2002 Inequalities from Complex Analysis, TCMM28

lgli/2002 Inequalities from Complex Analysis, TCMM28

American Mathematical Society

United States, United States of America

April 2002

0

lg1145988

{"isbns":["0883850338","9780883850336"],"publisher":"The Mathematical Association of America","series":"Carus Mathematical Monographs 28"}

Can be used as a text for a second course in complex analysis, a course on elementary functional analysis, or as supplement in courses in elementary Hilbert spaces or several complex variables.
<p>Inequalities from Complex Analysis is a careful, friendly exposition of the inequalities and positivity conditions for various mathematical objects arising in complex analysis. The author begins by defining the complex number field and discussing some standard mathematical analysis that leads up to recently published research on positivity conditions for functions of several complex variables. The development culminates in complete proofs of a stabilization theorum relating two natural positivity conditions for real-valued polynomials of several complex variables. The reader will also encounter the Bergman kernal funtion, the Fourier series, Hermitian linear algebra, the spectral theorem for compact Hermitian operators, plurisubharmonic functions, and some delightful inequalities. Numerous examples, exercises, and discussions of geometric reasoning are included to aid along the way. Undergraduate mathematics majors who have seen elecmentary real analysis can easily read the first five chapters of this book, and second year graduate students in mathematics can read the entire text. Some physicists and engineers may also find the topics and discussions useful. The inequalities and positivity conditions herein form the foundations for a small, but beautiful, part of complex analysis.</p>
<p>John P. D'Angelo was the 1999 winner of the Bergman Prize; he was cited for several important contributions to complex analysis, including his work on degenerate Levi forms and points of finite type, as well as work, some joint with David Catlin, on positivity conditions in complex analysis. D'Angelo received his Ph.D. from Princeton University. Since then he has held visiting professorial appointments at Princeton University, Washington University in St. Louis, the Institute for Advanced Study in Princeton, the Mathematical Sciences Research Institute in Berkeley, and the Mittag-Leffler Institute in Sweden. He has been named to the Incomplete List of Professors Ranked Excellent by their Students thirteen times and has served as a volunteer mathematics teacher in the Urabana schools</p>

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

Inequalities from Complex Analysis is a careful, friendly exposition of inequalities and positivity conditions for various mathematical objects arising in complex analysis. The author begins by defining the complex number field, and then discusses enough mathematical analysis to reach recently published research on positivity conditions for functions of several complex variables. The development culminates in complete proofs of a stabilization theorem relating two natural positivity conditions for real-valued polynomials of several complex variables. The reader will also encounter the Bergman kernel function, Fourier series, Hermitian linear algebra, the spectral theorem for compact Hermitian operators, plurisubharmonic functions, and some delightful inequalities. Numerous examples, exercises, and discussions of geometric reasoning appear along the way. Undergraduate mathematics majors who have seen elementary real analysis can easily read the first five chapters of this book, and second year graduate students in mathematics can read the entire text. Some physicists and engineers may also find the topics and discussions useful. The inequalities and positivity conditions herein form the foundation for a small but beautiful part of complex analysis. John P. D'Angelo was the 1999 winner of the Bergman Prize; he was cited for several important contributions to complex analysis, including his work on degenerate Levi forms and points of finite type, as well as work, some joint with David Catlin, on positivity conditions in complex analysis

Mathematical analysis requires a thorough study of inequalities.

2014-01-22