Characterized By Operator With Negative Coefficients Of New Subclasses Of Analytic Function And Its Properties

Complex analysis is the mathematics theory of studying on the complex functions,especially on meromorphic functions and analytic functions. It is one branch of mathematics with ancient and rich life, which is a classical research field and has attracted great attention of many mathematicians. The theories and methods of it is not used to solve many problems in differential equation、analytic number theory、differential geometry、topology, but also more generally used in many natural science areas, such as theoretical physics, aerodynamics, etc. Univalent functions and subordinate principle are the important contents of geometric function theory which include the theoretical study of area theorem、growth theorem、distortion theorem、 coefficient estimation、subordination chain、differential equation and differential subordination, etc. Many scholars have done a lot of work in this field, such as Miller and Mocanu[11].Since the1970s and1980s, with the application of convolution theory、differential subordination、integral operator, the research of complex analysis has already obtained series of achievements, many mathematicians in these respects did lots of work in these respects and also made many important results. Researchers based on analytic functions and applied convolution、 hyper-geometric functions to create many operators, and then did useful research jobs, such as Liu and Patel[8]、Cho, Kwon and Srivastava[5]、Sokol and Trojnar-Spelina[18]、Patel, Cho and Srivastava[15]. In recent years, many scholars have focused on the properties of the analytic functions with negative coefficients, such as Dziok and Srivastava[6], Liu and Srivastava[9][10]. H.M.Srivastava used operator Dλ to create the new subclasses of the analytic functions with negative coefficients, and studied the inclusion relations of the new functions. After that, A.Gangadharan still did hard work on neighborhoods properties of some functions associating with this operator, some special cases have been seemed as a consensus results.Under the inspiration of these articles, this paper defines an operator L(a,c) and discusses the relevant nature of operator, and the inclusion relations of the functions Sn(b)(γ,α,μ,β,a,c), Rn(b)(γ,α,γ,β,a,c),Mn(b)(γ,μ,β,a,c),Tn(b)(γ,μ,β,a,c). Applying the conclusions of this paper,one can be further research other properties of the analytic functions, especially starlike functions and convex functions.And this is also provides a theoretical and practical basis and research methods for functions. The following is the structure and main content of this paper:The first part is the preface:introduces the analytic functions with negative coefficients,(n,δ)-neighborhoods, convolution and concepts of L(a,c) operator.The second part is lemma:do some preparation for the third and the fourth part.The third part is some inclusion relations of (n,δ)-neighborhoods:this is one of the main conclusions of this paper. This part mainly discusses the relationships of neighborhoods of functions which defined in quotes.The fourth part is the inclusion relations:mainly discusses the inclusion relations of Sn(b)(γ,α,μ,β,a,c),Rn(b)(γ,α,γ,β,a,c),Mn(b)(γ,μ,β,a,c),Tn(b)(γ,μ,β,a,c). This is also the main conclusion of this paper.