期刊名称：Journal of Theoretical and Applied Information Technology
印刷版ISSN：1992-8645
电子版ISSN：1817-3195
出版年度：2018
卷号：96
期号：9
出版社：Journal of Theoretical and Applied
摘要：The complex analysis is one of the beautiful and important subject in the mathematics because of its applications which are not only in several aspects of the analysis, but even in several areas of the mathematics and science in general. Recently, much interesting has been given to investigate the differential inequalities which are containing the functions with their derivatives, and that field has been developed in the last sixty years. In the theory of the complex functions, the characterization of the given analytic function can be determined by using the differential inequality, and also by the differential inequality, most of the geometric properties can be described in the theory of geometric functions. In addition, one of the important branch in the field of the complex analysis is the geometric function theory because this theory deals with the geometric properties of analytic functions. Now days, in the geometric function theory, many applications have been existed for the theory of differential subordinations which its origins going back to Miller and Mocanu [38]. Then, the extensions of that theory can be found in different fields, such like the harmonic functions theory, the partial differential equations, the integral operators theory and the meromorphic functions theory. Due to that, the present paper had done in order to provide a literature study which focused on the theory of differential subordination and its applications in the study of analytic and univalent functions. Furthermore, the properties of the first, the second and the third order differential subordination are studied. This investigation concern on the most important results that have been introduced in the theory of differential subordination by the previous studies. Then, the contribution of this review is to provide an overview on the developments in the theory of differential subordination since the appearance of that concept right up to the recent years. The review of the previous studies may open research issues in the future works, and will allow the future researchers to have a useful background on the essential aspects of this research field.
