Properties of a New Subclass of Analytic Functions With Negative Coefficients De- fined by Using the Q-Derivative

In this paper we define a new class of analytic functions with negative coefficients involving the qdifferential operator. Our main purpose is to determine coefficient inequalities and distortion theorems for functions belonging to this class. Connections with previous results are pointed out.


Introduction
The quantum calculus or q−calculus has called the attention of many great researchers from Geometric function theory field due to its numerous applications in mathematics and physics. In 1999 was defined the first q-analogue of a starlike function by Ismail et al. [1]. In fact, many results obtained for univalent functions can be extended by using q-analogues functions. Recently, analytic functions with negative coefficient were studied in papers [5,[8][9][10].
Let U = {z : |z| < 1} be the open unit disk of the complex plane and let A represent the class of all functions of the form f (z) = z + ∞ ∑ n=2 a n z n , z ∈ U. (1.1) For 0 < q < 1, the q-derivative of a function f ∈ A is defined by (see [2]) with d q f (0) = f (0). Motivated by the aforementionated works, we define the following class of functions associated with Janowski functions: be the subclass of A consisting of functions of the form (1.1) and satisfying the condition Let T denote the subclass of analytic functions f ∈ A of the form n=2 a n z n , a n ≥ 0. (1.5) Also, we remark that ST (q, µ, k; A, B) reduces to the following known classes: (i) In case A = 1 − 2α, 0 ≤ α < 1, B = −1, µ = 1 and q → 1 − , we obtain the class S p (k, α) of k−uniformly starlike functions of order α(see [4] ); (ii) In case A = 1, B = −1, µ = 1, k = 1 and q → 1 − , we get the class S p of uniformly starlike functions (see [6]); (iii) In case µ = 1 and q → 1 − , we obtain the class k − ST [A, B] which was introduced and studied by Noor and Malik (see [3]); (iv) In case A = 1 − 2α, 0 ≤ α < 1, B = −1, µ = 1, k = 0 and q → 1 − , we get the class of starlike functions of order α denoted S * (α); (v) In case A = 1, B = −1, µ = 0, k = 0 and q → 1 − , we obtain the class R of functions whose derivative has positive real part.

Coefficient Estimates
We begin with a result that provides coefficient inequalities for functions in the class ST (q, µ, k; A, B).
Proof. It is suffices to prove that where the function g is defined by First of all, and (2.1) take place, then (k + 1)|g(z) − 1| is bounded above by 1. Hence f ∈ ST (q, µ, k; A, B) and the theorem is proved.
For f ∈ TST (q, µ, k; A, B) the converse of Theorem 2.1 is also true. The result is sharp with the extremal function f given by Proof. In view of Theorem 2.1, we need only to prove that (2.8) holds if f ∈ TST (q, µ, k; A, B). Conversely, assuming that f ∈ TST (q, µ, k; A, B) and z is real, we find that g(z) ≥ k|g(z) − 1|, where g is given in (2.3). Therefore, (2.10) Next, letting z → 1 − through real values, we obtain the inequality (2.8). Also, the equality in (2.8) take place for the function f given in (2.9). (n − α) · a n ≤ 1 − α.

2)
with equality for Proof. By using the hypothesis and the triangle inequality, we find that with equality for Corollary 3.2. (see [7], Theorem 4) Let the function f ∈ T given by (1.5) be in the class S * (α) ∩ T. Then, for |z| = r, with equality for The next theorem can be proven by employing similar techniques as in the demonstration of Theorem 3.1, so we will omit the details of our proof.
with equality for f given by (3.5).
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