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Overview
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.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.
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
Editorials
D. V. Feldman
"Takes the reader from the first properties of the complex numbers all the way to current research... Affords the undergraduate the pleasant opportunity to learn important basics by immediately seeing them fit together into something of beauty."—Choice
Harold P. Boas
"Suitable for self-study or for supplementary reading at the advanced undergraduate or beginning graduate level, this carefully crafted book discusses inequalities and positivity conditions related to functions of complex variables."—Mathematical Reviews
Jeffrey Nunemacher
"I really enjoyed reading this book...The first five chapters are accessible to the broad mathematical community with basic training in analysis and are useful for an honors course at the senior undergraduate level. The entire book offers an attractive but demanding introduction to modern complex analysis at the graduate level. "—MAA Online