Geometric Function Theory, Some Of The New Subclass

Since defined Ruscheweyh derivative of analytic functions by Hadamard product by S.Ruscheweyh[1], many scholars have studied classes of univalent or multivalent analytic functions associated with Ruscheweyh derivatives ([2-4]). Recently, based on different linear operators, some properties and characters of p-valent analytic functions and meromorphic functions have been investigated extensively([5-7]).There are mainly two chapters in this work. We deal with the properties and characters of the new subclasses in meromorphic and analytic functions in chapter One and chapter Two respectively.In the chapter One of this work, the meromorphic function is defined by Hypergeometric function LetΣ_p denote the class of mermorphic functions of the form Where * denotes Hadamard product or convolution.Firstly, making use of the linear operator H_p,q,s(α+1 ) and the principle of subordination between analytic functions, we introduce and investigate the following subclasses of themeromorphically p -valent function classΣ_p: Furthermore, some conclusions are acquired when we considerSome properties of classes are given in consideration of the integral operator . Also, we obtain the convolution properties ofSecondly, the class of meromorphic functions in , whose coefficient is positive, is investigated. The necessary and sufficient condition of f ( z ) falling into is obtained and the radius of starlike functions and convex functions are also considered.Finally, problems involving generalized neighborhoods and partial sums of meromorphic functions in classΣp are discussed, The relationship of theδ-neighborhoods of f ( z ) and the classes also considered.In the chapter Two of this work, a new subclass of analytic functions by Ruscheweyh derivatives is defined and some properties of are obtained. Let Ap be the class of functions which are analytic in the open unit disk U. A certain linear operatorThen Dm is Ruscheweyh derivative in cordition of m = ? p+ 1 and r = 1. Some properties of classis given. We also obtain the properties of Ap in connection with the operator .