Coefficient bounds for close-to-convex functions associated with vertical strip domain

Lemma 1. [4] Let f ∈A and α < 1 < β. Then f ∈S (α, β) if and only if zf ′ (z) f (z) ≺1 + β −α π i log

Lemma 1 means that the function fα,β : U →C defined by fα,β(z) = 1 + β −α π i log

Theorem 1. [4, Theorem 2.1] Let the function f ∈A be defined by (1.1). If f ∈S (α, β), then |an| ≤ nQ k=2 h k −2 + 2(β−α) π sin π(1−α) β−α i (n −1)! (n = 2, 3, . . .) . Here, in our present sequel to some of the aforecited works (especially [4]), we first introduce the following subclasses of analytic functions. Definition 2. Let α and β be real such that 0 ≤α < 1 < β. We denote by Sg (α, β) the class

Lemma 2. Let α, β and δ be real numbers such that 0 ≤α, δ < 1 < β and let the function f ∈A be defined by (1.1). Then f ∈Sg (α, β) if and only if zf ′ (z) g (z) ≺fα,β(z)

Lemma 3. Let the function g given by g (z) = ∞ X k=1 bkzk (z ∈U)

Theorem 2. Let α, β and δ be real numbers such that 0 ≤α, δ < 1 < β and let the function f ∈A be defined by (1.1). If f ∈Sg (α, β), then |an| ≤ nQ k=2 h k −2 + 2(β−δ) π sin π(1−δ) β−δ i n! +2 (β −α) nπ

Corollary 1. [5] Let α and δ be real numbers such that 0 ≤α, δ < 1 and let the function f ∈A be defined by (1.1). If f ∈C(α, δ), then |an| ≤2 (3 −2δ) (4 −2δ) · · · (n −2δ) n! [n (1 −α) + (α −δ)] (n = 2, 3, . . .) . Letting δ = 0, β →∞in Theorem 2, we have the following coefficient bounds for close-to- convex functions of order α.

Corollary 2. Let α be a real number such that 0 ≤α < 1 and let the function f ∈A be defined by (1.1). If f ∈C(α), then |an| ≤n (1 −α) + α (n = 2, 3, . . .) . Letting α = δ = 0, β →∞in Theorem 2, we have the well-known coefficient bounds for close-to-convex functions.

Corollary 3. [6] Let the function f ∈A be defined by (1.1). If f ∈C, then |an| ≤n (n = 2, 3, . . .) .

Theorem 3. Let α, β and δ be real numbers such that 0 ≤α, δ < 1 < β and let the function f ∈A be defined by (1.1). If f ∈Bg (α, β; ρ), then |an| ≤        nQ k=2 h k −2 + 2(β−δ)

Theorem 3. □ Letting β →∞in Theorem 3, we have the following consequence.

Corollary 4. Let α and δ be real numbers such that 0 ≤α, δ < 1 and let the function f ∈A be defined by (1.1). If f ∈Bg (α; ρ), then |an| ≤          nQ k=2 (k −2δ)

Corollary 5. Let α be a real number such that 0 ≤α < 1 and let the function f ∈A be defined by (1.1). If f ∈Hg (α; ρ), then |an| ≤[n (1 −α) + α] (1 + ρ) (2 + ρ) (n + ρ) (n + 1 + ρ) (n = 2, 3, . . .) , where ρ ∈R\ (−∞, −1] . Letting α = δ = 0, β →∞in Theorem 3, we have the following consequence.

Corollary 6. Let the function f ∈A be defined by (1.1). If f ∈Mg (ρ), then |an| ≤n (1 + ρ) (2 + ρ) (n + ρ) (n + 1 + ρ) (n = 2, 3, . . .) , where ρ ∈R\ (−∞, −1] . References [1] S. Bulut, Coefficient bounds for certain subclasses of close-to-convex functions of complex order, Filomat 31 (20) (2017), 6401–6408. [2] S. Bulut, M. Hussain and A. Ghafoor, On coefficient bounds of some new subfamilies of close-to-convex functions of complex order related to generalized differential operator, Asian-Eur. J. Ma