Some Properties of the Class of Univalent Functions with Negative Coefficients

The main object of this paper is to study some properties of certain subclass of analytic functions with negative coefficients defined by a linear operator in the open unit disc. These properties include the coefficient estimates, closure properties, distortion theorems and integral operators.


Introduction
which are analytic in the open unit disc.In particular,   For two functions   f z given by (1) and   : Let the function be given by:  , ; , ; , 0, 1, 2, 3, , where   k x denotes the Pochhammer symbol (or the shifted factorial) defined by: , .


Carlson and Shaffer [1] introduced a convolution operator on involving an incomplete beta function as: Our work here motivated by Catas [2], who introduced an operator on as follows: Now, using the Hadamard product (or convolution), the authors (cf.[3,4]) introduced the following linear operator: and   k x is the Pochhammer symbol.We defines a linear operator where and   k x the Pochhammer symbol .Special cases of this operator include: see [1]. the Catas drivative operator [2]:  the Ruscheweyh derivative operator [5] in the cases:  the Salagean derivative operator [6]: m D f z  the generalized Salagean derivative operator introduced by Al-Oboudi [7]: Let denote the class of functions which are analytic in the open unit disc.Following the earlier investigations by [8] and [9], we : : : denote the subclass of consisting of functions which satisfy We denote by consisting of all such functions [10].
The unification of the classes The class
Remark 1. 3 The class is a generalization of the following subclasses: defined and studied by [12]; ii)  and studied by [13] and [14]; Now, we shall use the same method by [17] to establish certain coefficient estimates relating to the new introduced class.

Coefficient Estimates Theorem 2.1 Let the function f be defined by (1). Then f belongs to the class if and only
, where Proof: Assume that the inequality (7) holds and let 1 z  .Then we have The result is sharp and the extremal functions are Consequently, by the maximum modulus theorem one obtains Conversely,suppose that Then from ( 6) we find that Choose values of on the real axis such that z which gives (7).Remark 2.2 In the special case Theorem 2.1 yields a result given earlier by [17]., , , , 1 1 and , , , 1 1 The equality in ( 10) and ( 11) is attained for the function f given by (9).
Proof: By using Theorem 2.2, we find from ( 6) that , which immediately yields the first assertion (10) of Theorem 2.3.
On the other hand, taking into account the inequality (6), we also have , , , , which, in view of the coefficient inequality (10), can be put in the form , , , , 1 1 and this completes the proof of (11).

Closure Theorem Theorem 3.1 Let the function
.
Hence by Theorem 2.1, also.

Distortion Theorems
and for z U  , where 0 i m   and is given by (8).

 
, , , , The equalities in (12) and ( 13) are attained for the function f given by Proof: Note that   , , , , , k By Theorem 2.2, we know that , , , , , 1 1 The assertions of ( 12) and ( 13) of Theorem 4.1 follow immediately.Finally, we note that the equalities ( 12) and ( 13) are attained for the function f defined by This completes the proof of Theorem 4.1.Remark 4.2 In the special case Theorem 4.1 yields a result given earlier by [17]. and for .The equalities in (15) and ( 16) are attained for the function z U  for z U  .The equalities in ( 17) and ( 18) are attained for the function 1 n f  given in (14).
The result is sharp with the extremal function 1 n f  given in (14)., , , ,

Theorem 4 . 1
Let the function f defined by (1) be in the class

Corollary 4 . 3
Let the function f defined by (1) be in the class

4
Let the function f defined by (1) be in the class

Corollary 4 . 5
Let the function f defined by (3) be in the class the unit disc is mapped onto a domain that contains the disc

Theorem 5 . 1
Let the function   f z defined by (1) be in the class