Child Analytic Functions Defined By Noor Integral Operator

Since defined Ruscheweyh derivative of analytic functions by S. Ruscheweyh[1], many scholars have studied classes of univalent or multivalent analytic functions associated with Ruscheweyh derivative. Several families of fractional operator, integral operator and derivative operator which are closely related with the Hadamard product (or convolution) were introduced and investigated in the context of univalent Function Theory. For instance, we choose to mention the Ruscheweyh derivative operator[1,2,19], the Noor integral operator[3,4], the Carlson-Shaffer operator[5,6], the Jung-Kim-Srivastava integral operator[7,8], the Dziok-Srivastava operator[9,10],and so on. Based on these different operators, some properties and characters of analytic functions and meromorphic functions have been studied extensively, For example, the inclusion relations, distortion and covering theorems,partial sums,convolution properties and so on. ([2],[12])In this paper, let A be the class of functions of the form which are analytic in the unit disk U = { z : z∈C且z <1}. For f ( z )∈A, n∈N , a certain operator I n: A�'A (called Noor integral operator) is defined as I n f ( z ) = f n(-1)( z )^* f ( z) such that and * denotes convolution or Hadamard product.Firstly, making use of Noor integral operator I n, a new subclass of analytic functions Q (α, n; A, B) is introduced in the open unit disk. As a generalized class of [2,21,22], three inclusion relations of it are obtained : Q (α, n; A, B ) (?) Q (α, n + 1; A, B); Q (α₂ , n; A, B ); Q (α₁, n; A, B); Q (α, n; A, B ) (?) Q (0, n;1 - 2ρ, -1).The consequence of the inclusion relations is accord with the conclusion of [18]. Secondly, the convolution properties of Q (α, n; A, B) are investigated, and some inclusion relations are preserved under the integral F_λ( f )( z). Finally, class Q^* (α, n; A, B) of analytic functions belonging to Q (α, n; A, B) with the negative coefficients is studied, the necessary and sufficient condition of f ( z ) falling into Q^* (α, n; A, B)is considered. Therefore the coefficient estimates, extreme points problems,the convexity and the radius of close-to-convex functions ,starlike functions, convex functions of class Q ? (α, n; A, B) are obtained.