Fekete-Szeg y Inequality for a Subclass of p-Valent Analytic Functions

which are analytic in the open unit disk E. Also, A1 = A, the usual class of analytic functions defined in the open unit disk E = {z : |z| < 1}. Let f(z) and g(z) be analytic in E. We say that the function f is subordinate to the function g and write f(z) ≺ g(z), if and only if there exists Schwarz function w, analytic in E such that w(0) = 0, |w(z)| < 1 for z ∈ E, and f(z) = g(w(z)). In particular, if g is univalent in E, then we have the following equivalence:

Definition 1.Let () be a univalent starlike function with respect to 1 which maps the unit disk  onto a region in the right half-plane which is symmetric with respect to the real axis with (0) = 1 and   (0) > 0. A function  ∈   is in the class  ,,, () if where 0 ≤  ≤ 1,  ≥ 0, and  > 0.
We have the following special cases.
We need the following results to obtain our main results.
for any complex number .The result is sharp for the functions () =  2 or () = .
Lemma 5 (see [7]).If  ∈ Ω, then for any real number  1 and  2 , the following sharp estimate holds: where The extremal function up to the rotations is of the form ) . ( The sets   ,  = 1, 2, . . ., 12 are defined as follows:
Proof.Since  ∈  ,,, (), therefore we have for a Schwarz function Now, Also, we have Comparing the coefficients of ,  2 ,  3 and after simple calculations, we obtain where  1 and  2 are defined in (25).It can be easily followed from (30) that where The results from (20) to ( 22) are obtained by using Lemma 3, (23) byusingLemma4, and (24) by using Lemma 5. To show that these results are sharp, we define the functions   (),   (), and   () such that It is clear that the functions   ,   ,   ∈  ,,, ().Let   : =  2 .If  <  1 or  >  2 , then the equality occurs for the function   or one of its rotations.For  1 <  <  2 , the equality is attained, if and only if  is  3 or one of its rotations.When  =  1 , then the equality holds for the function   or one of its rotations.If  =  2 , then the equality is obtained for the function   or one of its rotations.
The remaining proof of the theorem is similar to the proof of Theorem 6.