Turan P. On a new method of analysis and its applications

Wiley, 1984. — 600 p.
This book has a remarkable history. The present edition was announced twenty-five years before its publication. The first version of Turân's book was published in 1953 in Hungarian and in German. An improved Chinese edition followed in 1956. The theory was developed so rapidly that only a few years later these editions were out of date. In 1959 Turân announced a completely rewritten English edition. He wanted to cover all the theory and major applications of the power sum method which bears his name. A series of new inequalities and applications, found by Turân and other mathematicians, among whom Atkinson, de Bruijn, Cassels, Erdös, Gaier, Halâsz, Pintz, van der Poorten, Sós, Stark, and Uchiyama, led him to extend the manuscript and to rewrite old parts of the manuscript repeatedly. At his death on September 26,1976 he left a carefully organized manuscript comprising 57 sections. Sections 1-37 (except for Section 26) were in final form. For the remaining sections he had indicated the intended contents. These sections were written by Halâsz and Pintz following Turân's intentions as far as possible. Mrs. Sós took care of the further coordination. These efforts resulted in the publication of the English edition, eight years after Turân's death. The present book has 584 pages compared to 196 pages of the German edition. It is a unified treatise of the theory to which Turân devoted a considerable part of his life.