Subclass of k-uniformly starlike functions defined by symmetric q-derivative operator

Lemma 1.1. [6] If the function p ∈P, then 2B2 = B2 1 + x(4 −B2 1), 4B3 = B3 1 + 2(4 −B2 1)B1x −B1(4 −B2 1)x2 + 2(4 −B2 1)(1 −|x|2)z. for some x, z with |x| ≤1 and |z| ≤1. 2. Fundamental properties

Theorem 2.1. Let 0 < q < 1, and f ∈S be given by (1.5). If the inequality ∞ X n=2 hf [n]q(k + 1) −(k + α) i |an| ≤1 −α (2.1) holds true for some k (0 ≤k < ∞) and α (0 ≤α < 1) , then f ∈k-g ST q(α).

Corollary 2.1. Let 0 ≤k < ∞, 0 < q < 1, and 0 ≤α < 1. If, for f(z) = z + anzn, the following inequality |an| ≤ 1 −α f [n]q(k + 1) −(k + α) (n ≥2) holds, then f ∈k-g ST q(α). Specially f(z) = z + (1 −α)q q2(k + 1) + 1 −αz2 ∈k-g ST q(α).

Theorem 2.2. Let 0 ≤k < ∞, 0 < q < 1, and 0 ≤α < 1. A necessary and sufficient condition for f of the form f(z) = z −a2z2 −· · · (an ≥0) to be in the class k-g ST − q (α) is that ∞ X n=2 hf [n]q(k + 1) −(k + α) i an ≤1 −α. (2.2) The result is sharp, equality holds for the function f given by f(z) = z −

Theorem 2.3. Let 0 ≤k < ∞, 0 < q < 1 and 0 ≤α < 1. Let the function f defined by f(z) = z −a2z2 −· · · (an ≥0) be in the class k-g ST − q (α). Then for |z| = r < 1 it holds r − q (1 −α) (q2 + 1) (k + 1) −q(k + α)r2 ≤|f(z)| ≤r + q (1 −α) (q2 + 1) (k + 1) −q(k + α)r2. (2.4) Equality in (2.4) holds true for the function f given by f(z) = z + q (1 −α)

Theorem 2.4. Let 0 ≤k < ∞, 0 < q < 1 and 0 ≤α < 1. Let the function f with the Taylor series f(z) = z −a2z2 −· · · (an ≥0) be a member of the class k-g ST − q (α). Then for |z| = r < 1 1 − 2q (1 −α) (q2 + 1) (k + 1) −q(k + α)r ≤ |f ′(z)| ≤1 + 2q (1 −α) (q2 + 1) (k + 1) −q(k + α)r. (2.7)

Theorem 2.5. Let 0 ≤k < ∞, 0 < q < 1 and 0 ≤α < 1, and set f1(z) = z, fn(z) = z − 1 −α f [n]q(k + 1) −(k + α) zn (n = 2, 3, . . .). Then f ∈k-g ST − q (α) if, and only if, f can be expressed in the form f(z) = ∞ X n=1

Theorem 3.1. Let 0 ≤k < ∞, 0 < q < 1, 0 ≤α < 1, and let f ∈k-g ST q(α). 1. If U −P1q2(q2q4 −1) ≤0, V −P 2 1 q2 2q4 ≤0,

Lemma 1.1, and performing the necessary computations we obtain H2(2) = B4 N + M + P 2 1 q2 −q4S2 + 2P1q2q4S  + xB2(4 −B2)  M + 2P 2 1 q2 −2P1q2q4S  16q2 2q2 3q4 + −x2(4 −B2)

Corollary 3.1. Let q →1−. Then k-g ST q(α) →k-ST (α), for which P1 = 8 π2 . Then we get |a2a4 −a2 3| ≤16 π2 .

Theorem 3.2. Let 0 ≤k < ∞, 0 < q < 1, 0 ≤α < 1, and let f ∈k-g ST q(α). Then for complex µ it holds |a3 −µa2 2| ≤P 2 1 |q2 −µq3| + P2q2 2 q2 2q3 . In the case, when µ is real, then |a3 −µa2 2| ≤  

Corollary 3.2. Let 0 ≤k < ∞, 0 < q < 1, 0 ≤α < 1, and let f ∈k-g ST q(α). Then, the first Hankel determinant satisfy |a3 −a2 2| ≤q2(P2 + P 2 1 q) q4 + 1 − P 2 1 q2 q2 −q + 1.

Corollary 3.3. Under the assumption the same as in the Corollary 3.2 we have |a3| ≤q2P2 + P 2 1 q
q4 + 1 . 4. Acknowledgement The second author is supported by the Scientific and Technological Research Council of Turkey (TUBITAK 2214A). References [1] R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformaly convex functions and correspond- ding class of starlike functions, Tamkang J. Math. 28 (1997), 17-32. [2] L. C. Biedenharn, The quantum group SUq(2) and a q-analogue of the