MEROMORPHIC STARLIKE FUNCTIONS WITH RESPECT TO SYMMETRIC POINTS

The main purpose of this article is to introduce a class of meromorphic functions associated with the symmetric points in circular domain. We investigate the necessary and sufficient conditions, distortions theorem for this class. Furthermore, we obtain closure and convolutions properties, radii of starlikeness and partial sum results for these functions.


Introduction
Denoted by M, the class of functions f which are analytic in the D * = D {0} , where D = {z ∈ C : |z| < 1} and having the following series expansion form (1.1) f (z) = 1 z + ∞ n=1 a n z n , z ∈ D * .
For two functions f 1 (z) = 1 z + ∞ n=1 a n,1 z n and f 2 (z) = 1 z + ∞ n=1 a n,2 z n in D * the convolution or Hadamard product is defined by a n,1 a n,2 z n .

Coefficient Inequalities
This inequality is sharp.
Proof. Let us assume that condition ( Now for other part let suppose f ∈ MS * * [A, B] . We are to show that the inequality (2.1) , holds true.
Proof. It is easy to verify the relations To prove ( which implies that Using the relation (2.4), (2.6) become Conversly, suppose that the condition (2.3) hold, it follows that zf (z) = 0 for all z ∈ D.
If we denote Proof.
Then by equation δ i a n,i z n .
Proof. As Where we have used Theorem 2.1, on similar argument we have Thus prove the result.
Where we have used Theorem 2.1, and Thus prove the result. Proof. Since by Theorem 2.1, we have Using (1.1) along with some simple computation yields  Proof. Let f (z) be expressed in the form (2.12) , then and for above function, we have Hence proof is complete.

Partial Sums
Silverman [17] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. As a natural extension, one is interested to search results analogous to those of Silverman for meromorphic univalent functions. In this section, motivated essentially by the work of Silverman [17] and Cho and Owa [15]( also see [16,18]) we will investigate the ratio of a function of the form to its sequence of partial sums a n z n when the coefficients are sufficiently small to satisfy the condition analogous to For the sake of brevity we rewrite it as c n z n satisfies the inequality |ω(z)| ≤ |z|. Unless otherwise stated, we will assume that f is of the form (1.1) and its sequence of partial sums is denoted by f k (z) = 1 z + k n=1 a n z n .
The result (3.5) is sharp with the function given by Proof. Define the function w(z) by .
It suffices to show that |w(z)| ≤ 1. Now, from (3.8) we can write Hence we obtain From the condition (2.1), it is sufficient to show that which shows the bound (3.5) is the best possible for each k ∈ N.
We next determine bounds for f k (z)/f (z).
Authors contributions: All authors jointly worked on the results and they read and approved the final manuscript.