EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Theorem 1. Suppose f ∈Σ as defined by equation (4) belongs to the class. Then, the inequalities |a2| ≤ 2ε⟨κ; q⟩⟨2 −δ; q⟩ 𭟋

Theorem 2. Let f ∈Σ given by (4) belongs to the class BΣ(𭟋, λ, δ, B(κ)(ε, z; q)) and η ∈R. Then, we have a3 −ηa2 2 ≤          

Corollary 1. If f ∈Σ given by (4) belongs to the class BΣ(𭟋, 1, δ, B(κ)(ε, z; q)). Then |a2| ≤ ⟨2 −δ; q⟩ 𭟋⟨κ; q⟩ ε p 2 ⟨3 −δ; q⟩⟨κ; q⟩ε v u u u t Γq(3) ⟨3; q⟩2⟨2 −δ; q⟩(1 + ⟨2; q⟩) 𭟋⟨κ; q⟩2 −2⟨2; q⟩3 ⟨3 −δ; q⟩ ⟨κ; q2⟩+ ⟨κ; q⟩2

Corollary 2. Let f ∈Σ given by (4) belongs to the class BΣ(𭟋, 0, δ, B(κ)(ε, z; q)). Then |a2| ≤ 2 ⟨2 −δ; q⟩ 𭟋⟨κ; q⟩ ε p 2⟨3 −δ; q⟩⟨κ; q⟩ε s Γq(3) 4⟨3; q⟩2⟨2 −δ; q⟩𭟋⟨κ; q⟩2 −2⟨2; q⟩3 ⟨3 −δ; q⟩ ⟨κ; q2⟩+ ⟨κ; q⟩2
ε2 +⟨2; q⟩3 ⟨3 −δ; q⟩⟨κ; q2⟩
,

Definition 1. [19] The q-difference operator, also known as the q-derivative, is defined for a function f when 0 < q < 1 as follows: ðqf(z) =       

Definition 2. Let 𭟋∈C \ {0} and 0 ≤λ ≤1. A bi-univalent function f, defined in equation (4), belongs to the class BΣ(𭟋, λ, δ, B(κ)(ε, z; q)) if it satisfies the following conditions: 1 + 1 𭟋  ðq  ψδ qf(z)