Inclusion Properties for Certain Subclasses of Analytic Functions Associated with the Dziok-Srivastava Operator

Lemma 2.1 [17]. Let φ be convex univalent in U with φ(0) = 1 and Re{κφ(z) + ν} > 0 (κ,ν ∈C). If p is analytic in U with p(0) = 1, then p(z)+ zp′(z) κp(z)+ν ≺φ(z) (z ∈U) (2.1) implies p(z) ≺φ(z) (z ∈U). (2.2)

Lemma 2.2 [18]. Let φ be convex univalent in U and let ω be analytic in U with Re{ω(z)} ≥ 0. If p is analytic in U and p(0) = φ(0), then p(z)+ω(z)zp′(z) ≺φ(z) (z ∈U) (2.3) implies p(z) ≺φ(z) (z ∈U). (2.4)

Theorem 2.3. Let α1,λ > 1 and φ ∈. Then, λ+1,α1(q,s;η;φ) ⊂λ,α1(q,s;η;φ) ⊂λ,α1+1(q,s;η;φ). (2.5)

Theorem 2.4. Let α1,λ > 1 and φ ∈. Then, λ+1,α1(q,s;η;φ) ⊂λ,α1(q,s;η;φ) ⊂λ,α1+1(q,s;η;φ). (2.11)

Corollary 2.5. Let α1,λ > 1. Then, λ+1,α1(q,s;η;A,B) ⊂λ,α1(q,s;η;A,B) ⊂λ,α1+1(q,s;η;A,B), λ+1,α1(q,s;η;A,B) ⊂λ,α1(q,s;η;A,B) ⊂λ,α1+1(q,s;η;A,B). (2.14) Next, by using Lemma 2.2, we obtain the following inclusion relation for the class λ,α1(q,s;η,β;φ,ψ).

Theorem 2.6. Let α1,λ > 1 and φ,ψ ∈. Then, λ+1,α1(q,s;η,β;φ,ψ) ⊂λ,α1(q,s;η,β;φ,ψ) ⊂λ,α1+1(q,s;η,β;φ,ψ). (2.15)

Theorem 3.1. If f ∈λ,α1(q,s;η;φ), then Fc( f ) ∈λ,α1(q,s;η;φ) (c ≥0).

Theorem 3.2. If f ∈λ,α1(q,s;η;φ), then Fc( f ) ∈λ,α1(q,s;η;φ) (c ≥0).

Corollary 3.3. If f belongs to the class λ,α1(q,s;η;A,B) (or λ,α1(q,s;η;A,B)), then Fc( f ) belongs to the class λ,α1(q,s;η;A,B) (or λ,α1(q,s;η;A,B)) (c ≥0). Finally, we prove.

Theorem 3.4. If f ∈λ,α1(q,s;η,β;φ,ψ), then Fc( f ) ∈λ,α1(q,s;η,β;φ,ψ) (c ≥0).