The Properties Of Some Meromorphic Leafy Function

Geometric function theory is an important part of complex analysis, which mainly investigates geometrical properties of analytic function, and it is a mathematical field that geometry is related to analysis. It has a long history, whose origin can date back to the famous Riemann mapping theorem. In a long time of last century, many mathematicians, such as P.Koebe, L.Bieberbach, P.L.Duren and so on, made grate contribution to the development of geometric theory of complex variable function, therefore made its content very plenty and sound, and gained some perfect results. Although geometric function theory is a classic subject, it can be applied to many new fields, such as modern mathematical physics, fluid dynamics, particle differential equation.In recent years, many workers of complex analysis have aimed at p-valent analytic functions, and they have constructed a lot of operators by using Hadamard convolution, Hypergeometric function. 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 conclusion, such as Liu and Patel[4], Cho etc.[1], Sokol etc.[14], Patel and Srivastava[12]. Based on different linear operators, some properties and characters of meromorphic multivalent functions have also been investigated extensively, such as Liu[5], Liu and Srivastava[6].Motivated by their work, we define a operator Ipλ(a,c). Making use of the operator and differential subordination,∑a,cλ(η;p;h) are introduced. We investigate some useful properties of Ipλ(a,c) and the inclusion relationships of∑a,cλ(η;p;h).There are four parts in this article.The first part is introduction. We introduce p-valent meromorphic functions, differential subordination, the best dominant, Hadamard convolution, etc, and class∑a,cλ(η;p;h).The second part lists some important lemmas.The third part mainly investigates some related properties of the operator, related to differential subordination and estimation of argument.The fourth part mainly discuss the inclusion relationships of the class∑a,cλ(η;p;h).