Acta Universitatis Apulensis

Lemma 1. [26] Let p(z) ∈P and also let v be a complex number, then |c2 −vc2 1| ≤2 max {1, |2v −1|} , the result is sharp for functions given by p(z) = 1 + z2 1 −z2 , p(z) = 1 + z 1 −z .

Lemma 2. [26] Let p(z) ∈P, then |c2 −vc2 1| ≤      −4v + 2, if v ≤0; 2, if 0 ≤v ≤1; 4v −2,

Theorem 3. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 ̸= 0). If f(z) ∈Sq,b,λ j,k (φ), then |a3 −µa2 2| ≤ |B1b| 1 −λ + λ[3]q
[3]q −ψ3 max ( 1,

Theorem 4. Let φ(z) = 1 + B1z + B2z2 + · · · with B1 > 0. If f(z) given by (1) belongs to Cq,b,λ j,k (φ), then |a3 −µa2 2| ≤ |B1b| [3]q 1 −λ + λ[3]q
[3]q −ψ3
max ( 1,

Theorem 5. Let φ(z) = 1 + B1z + B2z2 + · · · with B1 > 0 and B2 ≥0. Let σ1 = [(1 −λ + λ[2]q) [2]q −ψ2] B2 1bψ2 + [(1 −λ + λ[2]q) [2]q −ψ2]2 (B2 −B1) [(1 −λ + λ[3]q) [3]q −ψ3] B2 1b , (21) σ2 = [(1 −λ + λ[2]q) [2]q −ψ2] B2 1bψ2 + [(1 −λ + λ[2]q) [2]q −ψ2]2 (B2 + B1) [(1 −λ + λ[3]q) [3]q −ψ3] B2 1b , (22) σ3 = [(1 −λ + λ[2]q) [2]q −ψ2] B2 1bψ2 + [(1 −λ + λ[2]q) [2]q −ψ2]2 B2

Theorem 6. Let φ(z) = 1 + B1z + B2z2 + · · · with B1 > 0 and B2 ≥0. Let χ1 = [2]2 q ((1 −λ + λ[2]q) [2]q −ψ2)  bB2 1ψ2 + ((1 −λ + λ[2]q) [2]q −ψ2) (B2 −B1)  B2 1b[3]q ((1 −λ + λ[3]q) [3]q −ψ3) , χ2 = [2]2 q ((1 −λ + λ[2]q) [2]q −ψ2)  bB2 1ψ2 + ((1 −λ + λ[2]q) [2]q −ψ2) (B2 + B1)

Corollary 7. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 ̸= 0). If f(z)given by (1) belongs to the class Sb,λ j,k (φ), then |a3 −µa2 2| ≤ |B1||b| 3 + 9λ −ψ3 max  1;

Corollary 8. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 ̸= 0). If f(z) given by (1)belongs to the class Cb,λ j,k (φ), then |a3−µa2 2| ≤ |B1||b| 9 + 18λ −3ψ3 max  1;

Corollary 9. Let φ(z) = 1 + B1z + B2z2 + · · · with B1 > 0 and B2 ≥0. Let σ4 = B2 1bψ2(2 + 4λ −ψ2) + (B2 −B1)(2 + 4λ −ψ2)2 B2 1b(3 + 9λ −ψ3) , σ5 = B2 1bψ2(2 + 4λ −ψ2) + (B2 + B1)(2 + 4λ −ψ2)2 B2 1b(3 + 9λ −ψ3) , σ6 = B2 1bψ2(2 + 4λ −ψ2) + B2(2 + 4λ −ψ2)2 B2 1b(3 + 9λ −ψ3)

Corollary 10. Let φ(z) = 1 + B1z + B2z2 + · · · with B1 > 0 and B2 ≥0. Let χ4 = 4(2 + 2λ −ψ2)[B2 1bψ2 + (B2 −B1)(2 + 2λ −ψ2)] B2 1b(9 + 18λ −3ψ3) , χ5 = 4(2 + 2λ −ψ2)[B2 1bψ2 + (B2 + B1)(2 + 2λ −ψ2)] B2 1b(9 + 18λ −3ψ3) , χ6 = 4(2 + 2λ −ψ2)[B2 1bψ2 + B2(2 + 2λ −ψ2)] B2 1b(9 + 18λ −3ψ3)