Some Applications Of Linear Operators And Differential Subordinations In Geometric Function Theory

Geometric Function Theory is an important and fruitful branch of classical complex analysis, which concentrate on geometric properties of various of analytic functions. It has a wide range of applications in some important branches of mathematics and other subject areas, and some new applications arise continually. In particular, an exciting new development associated with Geometric Function Theory is the Schramm-Loewner Evolution(SLE). In 1999, Combined Loewner differential equation from univalent func-tion theory with stochastic calculus, Oded Schramm established the definition of one-parameter family of SLE. SLE can be considered as a Brownian motion on the space of conformal maps, and it has important appications in both mathematics and statisti-cal physics. In addition, recently, with the applications of linear operators on analytic function space and differential subordination technique of analytic functions, Geometric Function Theory itself obtain a quick development. With the aid of linear operator or differential subordination, a lot of beautiful research results were obtained in Geometric Function theory recently, which concerning coefficient estimate, extreme points and close convex hull, starlikeness and univalence criteria, inclusion relationship and so on. This just shows that Linear operators and differential subordination become more powerful and effective tools in the study of the field.It is well know that differential subordination has close connections with differential inequalities, and a number of important classes of analytic functions were defined by differential inequality. In this Ph.D. thesis, we mainly consider and investigate some new and interesting properties of certain class of analytic functions by use techniques of linear operators and differential subordinations. This thesis is organized as follows.Chapter 1 is preface. The historical background, the subject, the recent develop-ment of geometric function theory and the main results and innovative contributions of this thesis are introduced.In Chapter 2, we introduce several families of linear integral operators. Combining the definition of four classical classes of analytic functions with Jung-Kim-Srivastava operator, we define four new function classes. By using Miller-Mocanu lemma, we obtain inclusion relationships of these classes, respectively. As their applications, we obtain corresponding results associated with Bernardi integral operator.In Chapter 3, by using convolution and Noor’s method, and with the aid of differ- ential operator Dn on meromorphic function space E, we define a new integral operator In,μ.Combining four classical meromorphic function classes with the new operator In,μ, we define four new meromorphic function classes. By using Miller-Mocanu lemma, we obtain inclusion relationships of these classes, respectively. In addition, by using Jack lemma, we obtain several integral preserving properties of integral operator Jc.In Chapter 4, we consider several Special subclasses of n-fold symmetric functions, such as starlike, convex, a-convex and a-quasi-convex functions with respect to n-fold symmetric points. In these subclasses, the a-quasi-convex functions with respect to n-fold symmetric points is new and defined by us first. Some interesting properties are provided for these subclasses. Firstly, for starlike and convex functions with respect to n-fold symmetric points, we obtain inclusion relationships, integral representations, convolution conditions, growth theorems, covering theorems and distortion theorems. Secondly, for a-convex and a-quasi-convex functions with respect to n-fold symmetric points, using convolution and differential subordination, we obtain inclusion relationships, integral representations, convolution conditions and integral properties.In Chapter 5, we introduce a new function expression and consider correspond-ing differential inequalities and first-order differential subordinations. By using Miller-Mocanu lemma and an useful differential subordination theorem, we obtain some new sufficient conditions for starlikeness and strongly star likeness.In Chapter 6, using Srivastava-Attiya operator, we define two new subclasses of k-fold symmetric functions. By using Herglotz formula and convolution technique, we obtain integral representations and related subordination results of these two subclasses. In addition, By using extremal theory which was developed by Brickman, Hallenbeck and Macgregor et al., corresponding properties of extreme points and close convex hull are provided. Finally, with the aid of subordination technique, we obtain a related subordination property.In Chapter 7, we introduce and investigate two new subclasses of meromorphic functions. Some interesting properties such as coefficient estimate, neighborhood prop-erty, partial sum property and inclusion relationships of these two subclasses are given.