Some Coefficient Inequalities of q-Starlike Functions Associated with Conic Domain Defined by q-Derivative

1Department of Mechanical Engineering, Sarhad University of Science and IT, Ring Road, Peshawar, Pakistan 2Department of Mathematics, COMSATS University Islamabad, Wah Campus, Pakistan 3Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada 4Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan


Introduction
Quantum calculus or q-calculus is none other than a version of classical calculus because in this, we do not take limits.We consider derivatives as differences whereas antiderivatives as sums.The q-derivative of a complex valued function , defined in the domain D, is given as follows.
provided that the function  is differentiable in domain D.
The function    has Maclaurin's series representation where For more details about q-derivatives, we refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14].We denote by A the class of functions () which are analytic in the open unit disc  = { : || < 1} and are of the form Let  denote the class of all functions in A which are univalent in .Also let  * and  be the subclasses of  consisting of all functions which map  onto a star shaped with respect to origin and convex domains, respectively.A function  is said to be subordinate to a function , written symbolically as  ≺ , if there exists a function  with (0) = 0, |()| < 1, such that () = (()) for  ∈ .
functions in compact disc.Several useful results related to the q-version of class of close to convex functions were proved by Sahoo and Sharma [16].Noor et al. [17] gave the research a new direction from application point of view and derived integral inequalities for relative harmonic preinvex functions.Very recently, many researchers of Geometric Function Theory like Noor et al. [18], Ramachandran et al. [19], Altinkaya et al. [3], Bulut [9], and Mahmood and Sokół [20] have contributed to the development of results in the background of q-calculus.The work on q-polynomials and (p,q)-polynomials also contributed remarkably to the field of q-calculus; see [10,11].

Preliminary Results
We need the following lemmas to prove our main results.Lemma 3 (see [30]).Let ℎ( Lemma 4 (see [31]).If () = 1+ 1 + 2  2 +⋅ ⋅ ⋅ is a function with positive real part in U, then, for any real number V, or one of its rotations.If V = 1, then, the equality holds if and only if () is reciprocal of one of the functions such that equality holds in the case of V = 0.Although the above upper bound is sharp, when 0 < V < 1, it can be improved as follows: and
Proof.Assuming that ( 21) holds, then it suffices to show that and we consider The last expression is bounded above by 1 if which reduces to and this completes the proof.
Theorem 7. Let the function  ∈  −   [, ] be of the form (5); then This result is sharp.
Proof.By definition, for  −   [, ], we have where If p () = 1 +    + ⋅ ⋅ ⋅ , then Now if () = 1 + ∑ ∞ =1     , then by ( 16) and ( 29), we get Now from (28), we have which implies that This implies that Comparison of coefficients of   gives us which reduces to For that, we use the principle of mathematical induction.For  = 2, we find from (37) that which results also from (27).Now for  = 3, we find from (37) that which also follows from (27).Now let inequality (38) be true for  = .We find from (37) that That is, which shows that inequality (38) is true for  =  + 1, and hence the required result.
For  = 0, The above result reduces to the following result, proved by Srivastava et al. [29].
For  = 0, the above result reduces to the following result, proved by Srivastava et al. [29].
For  → 1 − , Theorem 10 reduces to the following result.
Theorem 13.Let  ∈ 1 −   [, ], −1 ≤  ≤  ≤ 1 and be of the form (5). Then for real number , we have where 6 ( − ) (1 + ) in the proof of Theorem 10 and following the similar method as followed in Theorem 10 will lead us to the required result.
For  → 1 − , the theorem reduces to the following form.
Now we consider the inverse function F, defined as F() = F(()) = ,  ∈ U, and we find the following coefficient bound for inverse functions.