The book is nicely organized and well written for readers who have a background in real analysis and complex variable theory. The book is carefully prepared...It will be a valuable resource for research libraries. — Richard Chechile, *Journal of Mathematical Psychology*

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. — Jeffrey Nunemacher, *MAA Online*

This short book takes readers from the first properties of the complex numbers, all the way to current research. On the way, the readers 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....such a book affords the undergraduate the pleasant opportunity to learn important basics by immediately seeing them fit together into something of beauty. — D. V. Felman, *CHOICE*

**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.

Electronic ISBN: 9780883859704