On the Fekete-Szegö like inequality for meromorphic functions with fixed Residue d

In the present paper, we will consider the class of meromorphic starlike functions with fixed residue .   Silverman et al. (2008) has obtained sharp upper bounds for Fekete-Szego like functional  for certain subclasses of meromorphic functions. In this paper, we will find sharp upper bounds for  for the class meromorphic starlike functions with fixed residue . The aim of the present paper, is to completely solve Fekete Szego problem for a certain subclass of meromorphic starlike functions with fixed residue d. 
 
   
 
 Key words: Fekete-Szego inequality, starlike function, analytic function, subordination, meromorphic function.


Let
( ) H U be the set of functions which are regular in the unit disc and S denote the class of functions of the form 2 ( ) that are analytic and univalent in unit disc  Kanas and Ronning (1999) introduced the following classes: .
The class ( ) ST t is defined by the geometric property that the image of any circular arc centered at t is starlike with respect to ( ) f t and the corresponding class ( ) CV t is defined by the property that the image of any circular arc centered t is convex.*Corresponding author.E-mail: ozlemg@dicle.edu.tr.

Mathematical subject classification: 30C45.
Let ∑ denote the class of the functions of the form that are regular and univalent in with a simple pole at the origin with Residue 1.
For 0 1 t ≤ < , let t ∑ denote the class of functions f which are meromorphic and univalent in the unit disc U with the normalization lim ( ) with the topology given by uniform convergence on compact subsets of Then t A is locally convex linear topological space and t ∑ is a compact subset of t A (Schober, 1975).
In the punctured open unit disk and the class of all such meromorphic starlike functions in φ be an analytic funtions with positive real part on U with (0) 1 φ = , ' (0) 1 φ > , which maps the unit disk U onto a region starlike with respect to 1 and is symmetric with respect to the real axis.Let * ( ) φ where p denotes subordination between analytic functions.This class was studied by Silverman et al. (2008).They have obtained Fekete Szegö like inequality for functions in the class

Definition 1
Let ( ) z φ be an analytic funtions with positive real part on U with (0) 1 φ = , ' (0) 1 φ > , which maps the unit disk U onto a region starlike with respect to 1 and is symmetric with respect to the real axis.Let * ( ) where p denotes subordination between analytic functions and * ( ) To prove our result, we need the following lemmas.First lemma was obtained by Keogh and Merkes (1969).
The bounds are sharp.
, then there is a Schwarz function Now adding Equation ( 12) in (10), we have From this Equation (1), we obtain or equivalently Where The result of Equation ( 8) follows by an application of Lemma 1.If Clearly, the functions 1 ( ) F z , 2 ( ) ., we obtain the sharp inequality, Putting 1 d = and 0 t → in Theorem 1, we get the following result obtained by Silverman et al. (2008).

∑
that are regular in U and satisfy ( ) 1 p t = and ( ) Re ( ) 0 p z > for z U ∈ .