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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. 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