Hankel determinant problem of a subclass of

Lemma 1.1. Let f(z) ∈A . Let the qth Hankel determinant of f (z) for q ≥1, n ≥1 be defined by (1.4). Then, writting Δj(n) = Δj(n, z1, f(z)), we have Hq(n) =  2q−2(n) 2q−3(n + 1) · · · q−1(n + q −1) 2q−3(n + 1) 2q−4(n + 2) · · · q−2(n + q −2) ... ... ... ... q−1(n + q −1) q−2(n + q −2) · · · q(n + 2q −2)

Lemma 1.2. With z1 = n n+1y and v ≥0 any integer, j(n + v, z1, zf ′(z)) = j

Lemma 1.3. Let h1(z) be starlike univalent function in ℰ. Then (i) there exists a z1 with |z1| = r such that for all z, |z| = r | z −z1 | | h1(z) | ≤ 2r2 1 −r2 , see [25] (ii) r (1 + r)2 ≤| h1(z) |≤ r (1 + r)2 , see[26]. 2 Hankel determinant problem

Theorem 2.1. Let f(z) ∈˜Bk(λ, σ, β, γ )with 0 <b < 2 and let the qth Hankel determi- nant Hq(n) of f(z) be defined as in (1.4). Then Hq(n) = O (1) (M(r))1−γ ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ n γ ⎛ ⎝k 2 +1 ⎞ ⎠(1−σ)cos2λ+β−2 , q = 1

Lemma 1.1 to have Hq(n) = O(1) (M(r))1−γ n $ γ  k 2 +1  (1−σ)cos2λ+β−1 % q−q2 , k ≥ 8(q −1) (1 −σ)γ cos2λ −2 Arif et al. Journal of Inequalities and Applications 2012, 2012:22$

Corollary 2.1. For l = 0, b = 1, s = 0, f(z) ∈Bk(γ ), where the class Bk(γ ) was introduced by Noor et al. [14] and Hq(n) = O(1)(M(r))1−γ ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ n γ  k 2 +1  −1 , q = 1 n

Corollary 2.2. For λ = 0, γ = 1, f(z) ∈˜Tk(β, σ) and Hq(n) = O(1) ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ n (1−σ)  k 2 +1  +β−2, q = 1 n $ (1−σ)$

Definition 1.1. Let f(z) ∈A in E. Then f(z) ∈˜Bk(λ, σ, β, γ ), if for k ≥2, 0 ≤b ≤ 1, 0 ≤g ≤1, l is real with | λ |< π 2 there exists a function f1(z) ∈Vλ k (σ), 0 ≤s < 1, such that arg z1−γ f ′(z) f(z)  f(z) f ′

Definition 1.2. Let f(z) ∈A . Then the qth Hankel determinant of f (z) is defined for q ≥1, n ≥1 by Hq(n) =  an an+1 · · · an+q−1 an+1 an+2 · · · an+q−2 ... ...