Meromorphically Multivalent Functions Related To The Linear Operator In A Subclass

Geometric function of complex variable function, also known as the geometric theory, is ancient and rich life, one branch of mathematics is a classic area of research, has ever attracted great attention of many mathematicians. The theories and methods of it is not only can be used to solve the issues raised by differential equations, analytic number theory, differential geometry, topology and other branch of mathematics but also be more generally used in many natural science areas, such as theoretical physics, aerodynamics, etc. Univalent function and the principle of subordinate is one of the important contents of geometric function theory. Their theoretical studies, including the area theorem of univalent function, the growth theorem, deviation theorem, coefficient estimates, subordination chain, Briot-Bouquet differential equations and differential subordination, etc.. Since the seventies and eighties of the last century, with the application of convolution theory, differential subordination, fractional calculus operators and extreme points,the studying of geometric function theory also rejuvenated. Many mathematicians have done a lot of work about the convolution operator, differential subordination, combination of fractional calculus operators and univalent function theorem and they have got a lot of important results, such as Sanford S.Miller and Petru T.Mocanu[1].In recent years, many workers of complex analysis targeting the p-valent analytic functions, that is, their research extent has be widen from A1 to A p. Researchers have constructed a lot of operators by using Pochhammer symbols, Hadamard convolution, Hypergeometric function, the inverse of operator and so on, such as the Noor integral operator, Ruscheweyh derivative, Carlson-Shaffer operator and so on. They studied the properties of these operators, as well as the inclusion relationships and properties of the functions in the function class defined by the operator and achieved a number of important conclusions, such as Nak Eun Cho etc.[2], Xiu-Lian Fu and Ming-Sheng Liu[3], H.M.Srivastava and J.Patel[4], J.Patel etc.[5], M.K.Aouf etc.[6]. While noting the difference and contact between p-valent analytic functions and p-valent meromorphic functions, the researchers used a similar method to study the p-valent meromorphic functions. That is, they constructed a lot of operators by using Pochhammer symbols, Hadamard convolution, Hypergeometric function, the inverse of operator and so on. They studied the properties of these operators, as well as the inclusion relationships and properties of the functions in the function class defined by the operator and achieved a number of important conclusions, such as B.A.Vralegaddi and C.Somanatha[7], S.B.Joshi and H.M.Srivastava[8], Jin-Lin Liu and H.M.Srivastava[9], Jin-Lin Liu and H.M.Srivastava[10], M.K.Aouf[11].Motivated by [5], we define a operator I pλ( a , c) inΣp. Making use of operator I pλ( a , c) and differential subordination,∑λa , c (η; p ; A, B) are introduced. We will investigate the inclusion relationships of∑λa , c (η; p ; A, B) and some useful properties of I pλ( a , c).The followings are the construction and main contents of this paper: The first part is introduction. This part is prepared for the third and fourth part. We will introduce p-valent meromorphic functions , differential subordination, the best control, Hadamard convolution, etc and will gives some important definitions. The second part is a set of preliminary lemmas. We will introduce some preliminary lemmas will be used in this paper in this part.The third part is the inclusion relationships for the function class, which is one part of the main conclusions of this paper. In this part, we will investigate the inclusion relationships of the function class defined in the fist part of the paper.The fourth part is some related properties of the operator, which is also one part of the main conclusions of this paper. In this part, we will discuss some properties of the operator defined in the fist part, related to differential subordination, estimation of modulus and argument.