This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Few mathematical books are worth translating 50 years after original publication. Polyá-Szegö is one! It was published in German in 1924, and its English edition was widely acclaimed when it appeared in 1972. In the past, more of the leading mathematicians proposed and solved problems than today. Their collection of the best in analysis is a heritage of lasting value.

Pólya and Szegö’s classic treatment of isoperimetric inequalities in mathematical physics from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.

Papers on the location and behavior of zeros, including some of Polya's most influential work.The papers in this volume cluster about the following topics: the location of the zeros of polynomials and other analytic functions; the approximation of analytic functions by polynomials in which the location of zero is restricted (these papers represent some of Polya's most influential work); the behavior of the zeros of successive derivatives; the zeros of functions defined by trigonometric integrals; and the signs of derivatives and their analytic character. The volume also includes a paper that is not about zeros but contains a representation theorem for positive polynomials in several variables.