Inequalities from complex analysis /

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