Distortion Theorems in Relation to Linear Integral Operators

The present monograph consists of two parts. Before Part I, a chapter of introduction is supplemented, where an overview of the whole volume is given for reader's convenience. The former part is devoted mainly to expose linear inte- gral operators introduced by the author. Several properties of the operators are established, and specializations as well as generalizations are attempted variously in order to make use them in the latter part. As compared with the former part, the latter part is de- voted mainly to develop several kinds of distortions under actions of integral operators for various familiar function- also absolute modulus. real part. range. length and area. an- gular derivative, etc. Besides them, distortions on the class of univalent functions and its subclasses, Caratheodory class as well as distortions by a differential operator are dealt with. Related differential operators play also active roles. Many illustrative examples will be inserted in order to help understanding of the general statements. The basic materials in this monograph are taken from a series of researches performed by the author himself chiefly in the past two decades. While the themes of the papers pub- lished hitherto are necessarily not arranged chronologically Preface viii and systematically, the author makes here an effort to ar- range them as ,orderly as possible. In attaching the import- ance of the self-containedness to the book, some of unfamil- iar subjects will also be inserted and, moreover, be wholly accompanied by their respective proofs, though unrelated they may be.