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Abstract
In this paper we introduce and investigate two new subclasses of the function class $Σ$ of bi-univalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, and satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-MacLaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses. Several known or new consequences of these results are also pointed out.
Results & Lemmas (7)
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Lemma 2.1. · coeff
Lemma 2.1. [8, Lemma 2.1] Let the function Φ(z) = 1+ ∞ P n=1 hnzn, z ∈U, such that Φ ∈Pm(β). Then, |hn| ≤m(1 −β), n ≥1. By employing the…
Lemma 2.1. [8, Lemma 2.1] Let the function Φ(z) = 1+ ∞ P n=1 hnzn, z ∈U, such that Φ ∈Pm(β). Then, |hn| ≤m(1 −β), n ≥1. By employing the techniques used earlier by Deniz [4], in the following section we find esti- mates of the coefficients |a2| and |a3| for functions of the above-defined subclasses Sa,b,c Σ (γ, λ, β) and Ka,b,c Σ (γ, λ, β) of the function class Σ. 3. Coefficient Bounds for the Function Class Sa,b,c
Theorem 3.1.
Theorem 3.1. If the function f given by (1.1) belongs to the class Sa,b,c Σ (γ, λ, β), then |a2| ≤min (s m|γ|(1 −β) |(λ2 −2λ)ϕ2 2 + (3…
Theorem 3.1. If the function f given by (1.1) belongs to the class Sa,b,c Σ (γ, λ, β), then |a2| ≤min (s m|γ|(1 −β) |(λ2 −2λ)ϕ2 2 + (3 −λ)ϕ3|; m|γ|(1 −β) (2 −λ)ϕ2 ) (3.5) and |a3| ≤min m|γ|(1 −β) (3 −λ)ϕ3
Corollary 3.1.
Corollary 3.1. If the function f given by (1.1) belongs to the class Sa,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) |2ϕ3 −ϕ2 2|; m|γ|(1 −β)…
Corollary 3.1. If the function f given by (1.1) belongs to the class Sa,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) |2ϕ3 −ϕ2 2| ; m|γ|(1 −β) ϕ2 ) and |a3| ≤min m|γ|(1 −β) |2ϕ3 −ϕ2 2| + m|γ|(1 −β)
Corollary 3.2.
Corollary 3.2. If the function f given by (1.1) belongs to the class Ga,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) 3ϕ3; m|γ|(1 −β) 2ϕ2 )…
Corollary 3.2. If the function f given by (1.1) belongs to the class Ga,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) 3ϕ3 ; m|γ|(1 −β) 2ϕ2 ) and |a3| ≤m|γ|(1 −β) 3ϕ3 , where ϕ2 and ϕ3 are given by (1.5).
Theorem 4.1.
Theorem 4.1. If the function f given by (1.1) belongs to the class Ka,b,c Σ (γ, λ, β), then |a2| ≤min (s m|γ|(1 −β) |4(λ2 −2λ)ϕ2 2 + 3(3…
Theorem 4.1. If the function f given by (1.1) belongs to the class Ka,b,c Σ (γ, λ, β), then |a2| ≤min (s m|γ|(1 −β) |4(λ2 −2λ)ϕ2 2 + 3(3 −λ)ϕ3|; m|γ|(1 −β) 2(2 −λ)ϕ2 ) (4.1) and |a3| ≤min m|γ|(1 −β) 3(3 −λ)ϕ3
Corollary 4.1.
Corollary 4.1. If the function f given by (1.1) belongs to the class Ka,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) |6ϕ3 −4ϕ2 2|; m|γ|(1 −β)…
Corollary 4.1. If the function f given by (1.1) belongs to the class Ka,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) |6ϕ3 −4ϕ2 2| ; m|γ|(1 −β) 2ϕ2 ) and |a3| ≤min m|γ|(1 −β) 6ϕ3
Corollary 4.2.
Corollary 4.2. If the function f given by (1.1) belongs to the class Qa,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) 9ϕ3; m|γ|(1 −β) 4ϕ2 )…
Corollary 4.2. If the function f given by (1.1) belongs to the class Qa,b,c Σ (γ, β), then |a2| ≤min (s m|γ|(1 −β) 9ϕ3 ; m|γ|(1 −β) 4ϕ2 ) and |a3| ≤m|γ|(1 −β) 9ϕ3 , where ϕ2 and ϕ3 are given by (1.5).
Function classes studied:
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