Some Properties Of Subclasses Of Two Kinds Of Multi - Leaf Analytic Functions

Complex analysis is the mathematics theory of studying on the complex functions, especially on meromorphic functions and analytic functions. As a classical research field, the theories and methods in complex analysis can not only be used to solve the mathematical problems in analytic number theory, differential equation and differential geometry, etc., but is generally applied to many other natural science fields, such as theoretical physics and aerodynamics. As one important content of geometric function theory, univalent function and principle of subordination are primarily used to research such contents as the deviation theorem, different subordination, area theorem, coefficient estimate, differential equation, subordination chain and growth theorem of univalent function. Many scholars, such as Miller and Mocanu, etc., have done research into this aspect. Some other scholars have researched the subclasses of multivalen analytic functions which are multivalenly starlike with respect to origin, multivalently convex and multivalently close-to-convex in the open unit disk:in 1917, Lowner proposed a convex function derived from the concept of the bounded function in rotating boundary. Then, Paatero researched this kind of function at length. Those scholars, such as Pinchuk, Brannan, Kirwan and Padmanabhan, as well as Parvatham, Moulis, Coonce and Noor, once discussed boundary bounded functions and rotational radius bounded function.Many of the above scholars used the method of differential subordination and convolution (Hadamard product) in their study. Based on the subordination relationship defined by Schwarz function, Janowski and other scholars presented an introduction to many analytic function subclasses. In 1973, Rscheweyh and Sheil-Small used convolution method to verify Polya-Schoenberg conjecture. They proved that convex functions, star-like functions and close-to-convex functions were closed after convolution. Ruscheweyh et al. extended this concept. Many scholars used Ruscheweyh method and proved that some other analytic and convex (or other relevant) functions were also changeless after convolution. In addition to this, many interesting analytic function subclasses were derived by the use of Hadamard product or convolution in unit circle to define some operators, such as Carlson-Shaffer operator, Ruscheweyh derivative operator and Noor integral operator. Then, a systematic study was carried out on such natures as coefficient estimate, deviation theorem and inclusion relation of these function subclasses.Inspired by the above researches, this paper defined the two multivalent analytic functions Tλ,α,p(M, N) and Rk(n,p,β) in unit circle U by using differential subordination and Noor integral operator respectively, and discussed the properties of the two functions.The whole paper consists of four parts. Each of the components is given as follows:The first part presents an introduction to the basic concepts required for the research work in this paper, such as differential subordination, hypergeometric function and Noor integral operator, etc., and gives a definition to the two functions Tλ,α,p(M,N) and Rk (n,p,β). These play a key role in the main conclusions of this paper.The second part presents the relevant lemmas, to make preparations for the proofs in the third and fourth parts.The third part mainly discusses function class Tλ,α,p(M,N):discuss such natures as subordination relationship and radius problem in multivalent analytic function class Tλ,α,p(M, N) by using differential subordination relationship and inequality.The fourth part mainly discusses the inclusion relation of function class Rk ((n,p,β).