Hankel determinant for a class of analytic functions of complex order defined by convolution

Lemma 1 ([26]). Let (2.1) h(z) = 1 + ∞ X n=1 cnzn ≺1 + ∞ X n=1 Cnzn = H(z) (z ∈U). If the function H is univalent in U and H(U) is a convex set, then (2.2) |cn| ≤|C1| .

Lemma 2 ([10]). Let a function p ∈P be given by (2.3) p(z) = 1 + c1z + c2z2 + . . . (z ∈U), then, we have (2.4) |cn| ≤2 (n ∈N). The result is sharp.

Lemma 3 ([17, 18]). Let p ∈P be given by the power series (2.3), then for any complex number ν (2.5) c2 −νc2 1 ≤2 max{1; |2ν −1|}. The result is sharp for the functions given by p(z) = 1 + z2 1 −z2 and p(z) = 1 + z 1 −z (z ∈U).

Lemma 4 ([15]). Let a function p ∈P be given by the power series (2.3), then (2.6) 2c2 = c2 1 + κ(4 −c2 1) for some κ, |κ| ≤1, and (2.7) 4c3 = c3 1 + 2(4 −c2 1)c1κ −c1(4 −c2 1)κ2 + 2(4 −c2 1)  1 −|κ|2

Theorem 1. Let f(z) given by (1.1) belong to the class Sb γ (g(z); A, B), 0 ≤γ ≤1, −1 ≤B < A ≤1 and b ∈C∗, then (3.1) |ak| ≤ (A −B) |b| [1 + γ (k −1)] bk (k ∈N \ {1}) .

Theorem 2. Let f(z) ∈A, then a sufficient condition for f(z) to be in the class Sb γ (g(z); A, B) is (3.3) ∞ X k=2 [1 + γ(k −1)] bk |ak| ≤(A −B) |b| 1 + B . In the next two theorems, we obtain the result concerning Fekete–Szeg¨o inequality and an upper bound for the Hankel determinant for the class Sb γ (g(z); A, B).

Theorem 3. Let f(z) given by (1.1) belong to the class Sb γ (g(z); A, B), 0 ≤γ ≤1, −1 ≤B < A ≤1 and b ∈C∗, then (3.4) a3 −µa2 2 ≤(A −B) |b| (1 + 2γ) b3 · max ( 1, B + µbb3 (A −B) (1 + 2γ) (1 + γ)2 b2 2

Corollary 1. Let f(z) given by (1.1) belong to the class Sb γ(λ, ℓ, m, q, s, α1, β1; A, B), 0 ≤γ ≤1, −1 ≤B < A ≤1, m ∈N0, ℓ≥0, λ ≥0, q ≤s + 1, q, s ∈N0 and b ∈C∗, then (3.20) a3 −µa2 2 ≤ (A −B) (1 + ℓ)m |b| (1 + 2γ) (1 + ℓ+ 2λ)m Γ3(α1) × max ( 1, B + µb

Theorem 3, we obtain the following corollary.

Corollary 2. Let f(z) given by (1.1) belong to the class Sb γ (λ, ℓ, m; A, B), 0 ≤γ ≤1, −1 ≤B < A ≤1, m ∈N0, ℓ≥0, λ ≥0 and b ∈C∗, then (3.21) a3 −µa2 2 ≤(A −B) |b| (1 + 2γ) h 1+ℓ+2λ 1+ℓ im × max  

Corollary 3. Let f(z) given by (1.1) belong to the class Sγ [ρ, η, A, B, g(z)], 0 ≤γ ≤1, −1 ≤B < A ≤1 and b ∈C∗, then (3.22) a3 −µa2 2 ≤(A −B) (1 −ρ) cos η (1 + 2γ) b3 × max ( 1, B + µb3 (A −B) (1 + 2γ) (1 −ρ) e−iη cos η (1 + γ)2 b2 2

Theorem 4. Let f(z) given by (1.1) belong to the class Sb γ (g(z); A, B), 0 ≤γ ≤1, −1 ≤B < A ≤1 and b ∈C∗, then (3.23) a2a4 −a2 3 ≤(A −B)2 |b|2 (1 + 2γ)2 b2 3 .

Corollary 4. Let f(z) given by (1.1) belong to the class Sb γ(λ, ℓ, m, q, s, α1, β1; A, B), 0 ≤γ ≤1, −1 ≤B < A ≤1, m ∈N0, ℓ≥0, λ ≥0, q ≤s+1, q, s ∈N0 and b ∈C∗, then (3.32) a2a4 −a2 3 ≤ (A −B)2 |b|2 (1 + 2γ)2 h 1+ℓ+2λ 1+ℓ i2m Γ2 3(α1)

Theorem 4, we obtain the following corollary.

Corollary 5. Let f(z) given by (1.1) belong to the class Sb γ (λ, ℓ, m; A, B), 0 ≤γ ≤1, −1 ≤B < A ≤1, m ∈N0, ℓ≥0, λ ≥0 and b ∈C∗, then (3.33) a2a4 −a2 3 ≤ (A −B)2 |b|2 (1 + 2γ)2 h 1+ℓ 1+ℓ+2λ i2m . Putting b = (1 −ρ) e−iη cos η (|η| < π 2 , 0 ≤ρ < 1) in Theorem 4, we obtain the following corollary.

Corollary 6. Let f(z) given by (1.1) belong to the class Sγ [ρ, η, A, B, g(z)], 0 ≤γ ≤1, −1 ≤B < A ≤1 and b ∈C∗, then (3.34) a2a4 −a2 3 ≤(A −B)2 (1 −ρ)2 cos2 η (1 + 2γ)2 b2 3 . References [1] Abubaker, A., Darus, M., Hankel determinant for a class of analytic functions in- volving a generalized linear differential operator, Internat. J. Pure Appl. Math. 69 (3) (2011), 429–435. [2] Al-Oboudi, F. M., On univalent functions defined by a generalized Salagean operator, Internat. J. Math. Math. Sci. 27