FEKETE SZEGÖ PROPERTIES FOR THE CLASS OF MOCANU FUNCTIONS ASSOCIATED WITH q−RUSCHEWEYH OPERATOR

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The objective of this paper is to introduce and investigate new subclass of analytic functions involving q−derivative Ruscheweyh operator. For functions belonging to this class, we obtain coefficient estimates on Taylor Maclaurin series and the results on the famous Fekete Szegö inequality.


INTRODUCTION
The q-difference calculus or quantum calculus was initiated at the beginning of 19th century, that was initially developed by Jackson [8,9]. The q−calculus is one of the tool which is used to introduce and investigate many number of subclasses of analytic functions. Basic definitions and properties of q-difference calculus can be found in the book mentioned in [10]. The origin of fractional q-difference calculus has been found in the works by Al.Salam [3] and Agarwal [2]. Due to the application of q−calculus in various branches of science, recently, the area of q-calculus has attracted the serious attention of researchers. Later, geometrical interpretation of q−analysis has been recognised through studies on quantum groups. Mohammed and Darus [14] studied approximation and geometric properties of these q-operators for some subclasses of analytic functions in compact disk.
Let A denote the class of all functions f (z) of the form: which are analytic in the open unit disc U = {z ∈ C : |z| < 1}.
Let S be the subclass of A consisting of all univalent functions in U.
If f (z) and g(z) are analytic in U, then we say that the function f (z) is subordinate to g(z), if there exists a Schwarz function w(z), analytic in U with We denote this subordination by In particular, if the function g(z) is univalent in U, the above subordination is equivalence to A q-analog of the class of starlike functions was first introduced in 1990 [7] by means of the q-difference operator D q f (z) acting on functions f ∈ A given by (1.1) and 0 < q < 1, the q-derivative of a function f (z) is defined by (see [8,9]) .
For a function h(z) = z k , we observe that, where h is the ordinary derivative.
As a right inverse, Jackson [9] introduced the q-integral provided that the series converges. For a function h(z) = z k , we have where z 0 h(t)dt is the ordinary integral. Note that the q-difference operator plays an important role in the theory of hypergeometric series and quantum physics (see for instance [4,5,6,12,16]). Kanas and Rǎducanu in [11] used the Ruscheweyh q-differential operator to introduce and study some properties of (q, k) uniformly starlike functions of order α. One can clearly see that D q f (z) → f (z) as q → 1 − . This difference operator helps us to generalize the class of starlike functions S * analytically.
Ma and Minda [13] unified various subclasses of starlike and convex functions for which ei- is subordinate to a more general superordinate function.
For this purpose, they considered an analytic function φ (z) with positive real part in the unit disc U, with φ (0) = 1, φ (0) > 0 and φ maps U onto a region starlike, with respect to the real axis.
The classes of Ma-Minda starlike functions consists of functions f (z) ∈ A satisfying the sub-

PRELIMINARIES
In this session, we present some of the known concepts and new definitions defined in the open unit disc U.
Quite recently, Abdullah and Darus in [1] introduced the new differential operator D m,v q,µ,δ ,k,λ by For different values of v, k, λ , δ , β and µ, we get various differential operators explained as Remark in [1].
By inspiring the works of Abdullah and Darus [1], we now define the new subclass M m,v q (φ ) of A associated with the differential operator(2.1).
3. Let f ∈ A and 0 ≤ α ≤ 1, then f is said to be in the class M m,v q (φ ) if it satisfies the following subordination condition: In order to prove the main result, we need the following lemma.
the result is sharp for the function p(z) = 1 + z 2 1 − z 2 and p(z) = we also need the following results for our investigation.

MAIN RESULTS
The main purpose of this paper is to obtain the Fekete-Szegö inequality for certain class of analytic functions defined by the differential operator involving q−Ruscheweyh operator.
Proof. Observe that the condition in (2.2) can be written as follows: Here, the function ω(z) is analytic in U with the condition ω(0) = 0 and |ω(z)| < 1 in U.
Let h(z) be an analytic function defined in U with ℜ{h(z)} > 0 and h(0) = 1 be given by c n z n for z ∈ U, then Since ω(z) is a Schwarz function, we have

Upon computation we get,
and (3.7) Therefore, We get our desired result by applying Lemma 2.4. This completes the proof of Theorem 3.1.
Remark 3.2. If we set α = 0 and α = 1, then we have the results of Theorem 5 and Theorem 6 obtained by Abdullah and Darus [1] respectively.
If we set m = 0 and v = 0 in Theorem 3.1, we thus obtain the following: belongs to M q (φ ), then The result is sharp.
Proof. The Proof is followed by Lemma 2.5.