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**Differential subordinations: theory and applications.**
*(English)*
Zbl 0954.34003

Pure and Applied Mathematics, Marcel Dekker. 225. New York, NY: Marcel Dekker. xi, 459 p. (2000).

A differential subordination in the complex plane is the generalization of a differential inequality on the real line. It is the authors who first introduced the concept “differential subordination” in a 1981 paper [Mich. Math. J. 28, 157-171 (1981; Zbl 0456.30022)]. Since then more than 300 articles related to this topic have appeared. This monogragh describes for the first time in book form the basic theory and a multitude of applications in the study of differential subordinations.

This book contains eight chapters and over 400 bibliographic items. The first chapter presents concise preliminaries. In Chapters 2, 3 and 4 the authors discuss the basic theory and various applications of first-order and second-order differential subordinations. In Chapter 5 several special differential subordinations are considered. The third-order and the Euler Nth-order differential subordinations are studied in Chapter 6. One can see from these chapters that differential subordination is also a useful technique for studying geometric function theory. Many proofs of known theorems are simplified and many results are extended. Chapter 7 presents an extension of differential subordination in the plane to several complex variables. The final chapter reviews applications of differential subordination in other fields, including harmonic functions, meromorphic univalent functions, functions analytic in the upper half plane and holomorphic vector-valued functions.

The Jack-Miller-Mocanu lemma and its extensions are the basis for most of the results in the book. The advanced technique LĂ¶wner chain plays an important role in the proofs of many results in the text.

This book contains eight chapters and over 400 bibliographic items. The first chapter presents concise preliminaries. In Chapters 2, 3 and 4 the authors discuss the basic theory and various applications of first-order and second-order differential subordinations. In Chapter 5 several special differential subordinations are considered. The third-order and the Euler Nth-order differential subordinations are studied in Chapter 6. One can see from these chapters that differential subordination is also a useful technique for studying geometric function theory. Many proofs of known theorems are simplified and many results are extended. Chapter 7 presents an extension of differential subordination in the plane to several complex variables. The final chapter reviews applications of differential subordination in other fields, including harmonic functions, meromorphic univalent functions, functions analytic in the upper half plane and holomorphic vector-valued functions.

The Jack-Miller-Mocanu lemma and its extensions are the basis for most of the results in the book. The advanced technique LĂ¶wner chain plays an important role in the proofs of many results in the text.

Reviewer: Liquan Liu (Harbin)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34Mxx | Ordinary differential equations in the complex domain |