Certain Bound for q-Starlike and q-Convex Functions with respect to Symmetric Points

The q-difference calculus or quantum calculus was initiated at the beginning of 19th century that was initially developed by Jackson [1, 2]. Basic definitions and properties of q-deference calculus can be found in the book mentioned in [3]. The fractional q-difference calculus had its origin in the works by Al-Salam [4] and Agarwal [5]. Recently, the area of qcalculus has attracted the serious attention of researchers.The great interest is due to its application in various branches of mathematics and physics. Mohammed and Darus [6] studied approximation and geometric properties of these q-operators for some subclasses of analytic functions in compact disk. LetA denote the class of all functions f(z) of the form


Introduction
The -difference calculus or quantum calculus was initiated at the beginning of 19th century that was initially developed by Jackson [1,2].Basic definitions and properties of -deference calculus can be found in the book mentioned in [3].The fractional -difference calculus had its origin in the works by Al-Salam [4] and Agarwal [5].Recently, the area of calculus has attracted the serious attention of researchers.The great interest is due to its application in various branches of mathematics and physics.Mohammed and Darus [6] studied approximation and geometric properties of these -operators for some subclasses of analytic functions in compact disk.
Let A denote the class of all functions () of the form which are analytic in the open unit disk U = { ∈ C : || < 1}.
Let S be the subclass of A consisting of all univalent functions in U.
If () and () are analytic in U, then we say that the function () is subordinate to (), if there exists a Schwarz function (), analytic in U with such that  () =  ( ()) ( ∈ U) .
We denote this subordination by
Ma and Minda [15] unified various subclasses of starlike and convex functions for which either quantity   ()/() or quantity 1 +   ()/  () is subordinate to a more general superordinate function.For this purpose, they considered an analytic function  with positive real part in the unit disc U, with (0) = 1,   (0) > 0, and  maps U onto a region starlike, with respect to the real axis.
Sakaguchi [16] introduced and studied class  *  of starlike functions with respect to symmetric points.Class  * , () defined below is the generalization of class  *  .
The main objective of the present paper is to derive the Fekete-Szegö inequality for functions in classes  * , () and  , ().Also, by using convolution product, we give the applications of our results to the defined functions and in particular we consider classes   , () and   , () defined by fractional derivatives.
In order to prove the main result, we need the following lemma.
When ] < 0 or ] > 1, the equality holds if and only if () = (1 + )/(1 − ) or one of its rotations.When 0 < ] < 1, then the equality holds if and only if () = (1 +  2 )/(1 −  2 ) or one of its rotations.If ] = 0, the equality holds if and only if or one of its rotations.While ] = 1, equality holds if and only if () is the reciprocal of one of the functions for which the equality holds in the case of ] = 0. Also the above upper bound can be improved as follows: when International Journal of Mathematics and Mathematical Sciences 3 We also need the following results in our investigation.

Main Results
Theorem 5.
If () given by ( 1) belongs to  * , (), then where ) , ) . ( The result is sharp. Proof.For  ∈  * , (), let From ( 18), we obtain Since () is univalent and  ≺ , the function is analytic and has positive real part in U. Thus we have and from ( 18) and (21), where Our result now follows by an application of Lemma 3. To show that these bounds are sharp, we define the functions   ( = 2, 3, . ..) by and the functions   and   (0 ≤  ≤ 1) by Clearly the functions   ,   ,   ∈  * , ().Also we write   :=  2 .If  <  1 or  >  2 , the equality holds if and only if  is   or one of its rotations.When  1 <  <  2 , the equality holds if and only if  is  3 or one of its rotations.If  =  1 then the equality holds if and only if  is   or one of its rotations.If  =  2 then the equality holds if and only if  is   or one of its rotations.

Applications to Functions Defined by Fractional Derivative
In order to introduce classes   , () and   , () we need the following.
Definition 11 (see [20]).Let () be analytic in a simply connected region of the -plane containing the origin.The fractional derivative of  of order  is defined by where the multiplicity of (−)   ) . ( The result is sharp.