Cover image for Inequalities from complex analysis
Title:
Inequalities from complex analysis
Personal Author:
Series:
Carus mathematical monographs ; 28
Publication Information:
Washington, D.C. : Mathematical Association of America, 2002
ISBN:
9780883850336

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30000010046803 QA331.7 D36 2002 Open Access Book Book
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Summary

Summary

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. ohn 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.


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Choice Review

Polynomials (and more generally power series) embody a curious duality that lies at the heart of many mathematical problems: one may see these objects either through their values or through their coefficients. For example, Hilbert's Seventeenth Problem asks one to understand the coefficients of those real polynomials that take only positive values. D'Angelo (Univ. of Illinois) treats recent work on analogous questions, but over the complex numbers, specifically the so-called stabilization of positive bihomogeneous polynomials. Remarkably, the author demands no advanced preparation beyond basic calculus. This short book takes readers from the first properties of the complex numbers all the way to current research. On the way, the reader will acquire essential tools from complex analysis, linear algebra, Hilbert space, several complex variables, Fourier analysis, and operator theory. Even more remarkably, the pace seems leisurely, with many delightful digressions, some nearly as interesting as the main results. Much as runners might prefer to take their exercise cross-country rather than tediously looping a track, such a book affords the undergraduate the pleasant opportunity to learn important basics by immediately seeing them fit together into something of beauty. Upper-division undergraduates through faculty. D. V. Feldman University of New Hampshire


Table of Contents

1 Complex numbers
2 Complex Euclidean spaces and Hilbert space
3 Complex analysis in several variables
4 Linear transformations and positivity conditions
5 Compact and integral operators
6 Positivity conditions for real-valued functions
7 Stabilisation for bihomogenous polynomials and applications