Starlike functions of complex order with bounded radius rotation by using quantum calculus

In the present paper, we study on the subclass of starlike functions of complex order with bounded radius rotation using q− difference operator denoted by Rk(q, b) where k ≥ 2, q ∈ (0, 1) and b ∈ C\{0}. We investigate coefficient inequality, distortion theorem and radius of starlikeness for the class Rk(q, b).

In 1909 and 1910, Jackson [5,6] initiated a study of q− difference operator by where B is a subset of complex plane C, called q− geometric set if qz ∈ B, whenever z ∈ B. Note that if a subset B of C is q− geometric, then it contains all geometric sequences {zq n } ∞ 0 , zq ∈ B. Obviously, D q f (z) → f (z) as q → 1 − .Note that such an operator plays an important role in the theory of hypergeometric series and quantum physics (see for instance [1,3,7]).
For a function f (z) = z n , we observe that Therefore we have where [n] q = 1−q n 1−q .Clearly, as q → 1 − , [n] q → n.Denote by P q the family of functions of the form p(z) = 1 + p 1 z + p 2 z 2 + p 3 z 3 + . . ., analytic in D and satisfying the condition where q ∈ (0, 1) is a fixed real number.

Lemma 1.1 ([2]
). p ∈ P q if and only if p(z) ≺ 1+z 1−qz .This result is sharp for the functions p(z , where φ is a Schwarz function. A function p analytic in D with p(0) = 1 is said to be in the class P k (q), k ≥ 2, q ∈ (0, 1) if and only if there exists p 2 (z).
For q → 1 − , P k (q) ≡ P k , (see [10]); for k = 2, q → 1 − , P k (q) ≡ P is the well known class of functions with positive real part.Also, for k = 2, P k (q) ≡ P q consists of all functions subordinate to 1+z 1−qz , z ∈ D. Definition 1.1.Let f of the form (1) be an element of A. If f satisfies the condition with k ≥ 2, q ∈ (0, 1), then f is called q− starlike function with bounded radius rotation denoted by R k (q).This class was introduced and studied by Noor et al. [9].
Definition 1.2.Let f of the form (1) be an element of A. If f satisfies the condition with k ≥ 2, q ∈ (0, 1), b ∈ C\{0}, then f is called q− starlike function of complex order with bounded radius rotation denoted by reduces to traditional class of the starlike functions S * .
We investigate coefficient inequality, distortion theorem and radius of starlikeness for the class R k (q, b).

Main Results
We first prove coefficient inequality for the class R k (q, b).For our main theorem, we need the following lemma.
This inequality is sharp for every n ≥ 2.
Proof.In view of definition of the class R k (q, b) and subordination principle, we can write where p ∈ P k (q) with p(0 Comparing the coefficients of z n on both sides, we obtain for all integer n ≥ 2. In view of Lemma 2.1, we get In order to prove (2), we will use process of iteration.Let c = k 2 |b|(1 + q) and use our assumption |a 1 | = 1 in (3), we obtain successively Hence induction shows that for n, we obtain This proves (2).This inequality is sharp, because extremal function is the solution of the q− differential equation This is well known coefficient inequality for starlike functions.
We now introduce distortion theorem and radius of q− starlikeness for the class R k (q, b).
This bound is sharp.
Proof.In view of Lemma 2.2 and subordination principle, we write Therefore, after routine calculations, we get After calculations in (5), we obtain ( 6) On the other hand, we have Considering ( 6) and ( 7) together, respectively, we get Taking q− integral on both sides of the last inequalities, we get (4).This bound is sharp, because extremal function is the solution of the q− differential equation Corollary 2.2.Taking q → 1 − and b = 1 in Theorem 2.2, we get the following well known result: 1 − z .
If q → 1 − , b = 1, then this radius reduces to r * (f . This is the radius of the class R k which was obtained by Pinchuk (see [10]).For q → 1 − , k = 2, we get the starlikennes of starlike functions of complex order