On Fekete-Szegö Problems for Certain Subclasses Defined by q-Derivative

in the open unit disk U = {z ∈ C : |z| < 1}. For two analytic functions f and g in U, the subordination between them is written as f ≺ g. Frankly, the function f(z) is subordinate to g(z) if there is a Schwarz function w with w(0) = 0, |w(z)| < 1, for all z ∈ U, such that f(z) = g(w(z)) for all z ∈ U. Note that, if g is univalent, then f ≺ g if and only if f(0) = g(0) and f(U) ⊆ g(U). In [1, 2], Jackson defined the q-derivative operator Dq of a function as follows: Dqf (z) = f (qz) − f (z) (q − 1) z (z ̸ = 0, q ̸ = 0) (2)


Introduction
Denote by A the class of all analytic functions of the form in the open unit disk U = { ∈ C : || < 1}.
Making use of the -derivative, we define the subclasses S *  () and C  () of the class A for 0 ≤  < 1 by () ) > ,  ∈ U} . ( These classes are also studied and introduced by Seoudy and Aouf [16]. where S * () and C() are, respectively, the classes of starlike of order  and convex of order  in U ( [17,18]).Next, we state the -analogue of Ruscheweyh operator given by Aldweby and Darus [8] that will be used throughout.
Definition 1 (see [8]).Let  ∈ A. Denote by R   the analogue of Ruscheweyh operator defined by where []  !given by is as follows: From the definition we observe that if  → 1, we have where R  is Ruscheweyh differential operator defined in [19].
Using the principle of subordination and -derivative, we define the classes of -starlike and -convex analytic functions as follows.
Definition 2. For  ∈  and  > −1, the class S * , () which consists of all analytic functions  ∈ A satisfies Definition 3.For  ∈  and  > −1, the class C , () which consists of all analytic functions  ∈ A satisfies To prove our results, we need the following.
The result is sharp given by Lemma 5 (see [18]).

Main Results
Now is our theorem using similar methods studied by Seoudy and Aouf in [16].
Proof.If  ∈ S * , (), then there is a function () in U with (0) = 0 and |()| < 1 in U such that Define the function () by Since () is a Schwarz function, immediately Re(()) > 0 and (0 Then from ( 16), (17), and ( 18), obtain Since we have From the last equation and ( 18), we obtain A simple computation in (18) and knowing that Then, from last equation and ( 18), we see that or equivalently, we have Therefore where By an application of Lemma 4, our result follows.Again by Lemma 4, the equality in ( 15) is gained for Thus Theorem 6 is complete.
Similarly, we can prove for the class C , ().We omit the proofs.