New Classes of Analytic Functions Involving Generalized Noor Integral Operator

Lemma 1.1. Let fz ∈A for all z ∈U then i I01, 11,1; 1, 1/n −11,pfz  fz. ii I11, 11,1; 1, 1/n −11,pfz  zf′z. iii zIλαj, Aj1,q; βj, Bj1,pfz′  λ1Iλ1αj, Aj1,q; βj, Bj1,pfz−λIλαj, Aj1,q; βj, Bj1,pfz. In the following definitions, we introduce new classes of analytic functions containing generalized Noor integral operator 1.15.

Lemma 1.6 see 8. Let qz be univalent in the unit disk U and θ and let φ be analytic in a domain D containing qU with φw / 0, when w ∈qU. Set Qz : zq′zφqz, hz : θqz  Qz. Suppose that 1 Qz is starlike univalent in U, 2 Rzh′z/Qz > 0 for z ∈U. If θpz  zp′zφpz ≺θqz  zq′zφqz, 1.24 then pz ≺qz, 1.25 and qz is the best dominant.

Lemma 1.7 9. Let qz be convex univalent in the unit disk U and let ϑ and ϕ be analytic in a domain D containing qU. Suppose that 1 zq′zϕqz is starlike univalent in U, 2 R{ϑ′qz/ϕqz} > 0 for z ∈U. If pz ∈Hq0, 1 ∩Q, with pU ⊆D and ϑpz  zp′zϕz being univalent in U and ϑqz  zq′zϕqz ≺ϑpz  zp′zϕpz, 1.26 then qz ≺pz, 1.27 and qz is the best subordinant.

Theorem 2.1. Let fz ∈A. Then fz ∈S μ λαj, Aj1,q; βj, Bj1,p if and only if ∞  n2 Hn−1 an μλ  1n−1 −λ  1n −λn  ≤1 −μ, 0 ≤μ < 1, 2.1 where Hn−1 : p

Corollary 2.2. Let the function fz belong to the class S μ λαj, Aj1,q; βj, Bj1,p. Then an  ≤ 1 −μ Hn−1 μλ  1n−1 −λ  1n −λn , 0 ≤μ < 1, 2.6 where Hn−1 is defined in 2.2.

Theorem 2.3. Let fz ∈A. Then fz ∈C μ λαj, Aj1,q; βj, Bj1,p if and only if ∞  n2 nHn−1 an μλ  1n−1 −λ  1n −λn  ≤1 −μ, 0 ≤μ < 1, 2.7 where Hn−1 is defined in 2.2.

Corollary 2.4. Let the function fz belong to the class C μ λαj, Aj1,q; βj, Bj1,p. Then an  ≤ 1 −μ nHn−1 μλ  1n−1 −λ  1n −λn , 0 ≤μ < 1, 2.8 where Hn−1 is defined in 2.2.

Theorem 2.5. Let fz ∈S μ λαj, Aj1,q; βj, Bj1,p, then fz  ≥|z| − 1 −μ H1 μλ  11 −λ  12 −λ2 |z|2, fz  ≤|z|  1 −μ H1 μλ  11 −λ  12 −λ2 |z|2,

Corollary 2.6. Under the hypothesis of Theorem 2.5, fz is included in a disk with its center at the origin and radius r given by r  1  1 −μ H1 μλ  11 −λ  12 −λ2 . 2.14 In the same way, we can prove the following result.

Theorem 2.7. Let fz ∈C μ λαj, Aj1,q; βj, Bj1,p then fz  ≥|z| − 1 −μ 2H1 μλ  11 −λ  12 −λ2 |z|2, fz  ≤|z|  1 −μ 2H1 μλ  11 −λ  12 −λ2 |z|2,

Corollary 2.8. Under the hypothesis of Theorem 2.7, fz is included in a disk with its center at the origin and radius r given by r  1  1 −μ 2H1 μλ  11 −λ  12 −λ2 . 2.16 We next study some properties of the classes S μ λαj, Aj1,q; βj, Bj1,p and C μ λαj, Aj1,q; βj, Bj1,p.

Theorem 2.9. Let λ > −1 and 0 ≤μ1 < μ2 < 1. Then S μ2 λ αj, Aj  1,q; βj, Bj  1,p  ⊂S μ1 λ αj, Aj  1,q; βj, Bj

Theorem 2.10. Let −1 < λ1 ≤λ2 and 0 ≤μ < 1. Then S μ λ1 αj, Aj  1,q; βj, Bj  1,p  ⊇S μ λ2 αj, Aj  1,q; βj, Bj

Theorem 2.11. Let λ > −1 and 0 ≤μ1 < μ2 < 1. Then C μ2 λ αj, Aj  1,q; βj, Bj  1,p  ⊂C μ1 λ αj, Aj  1,q; βj, Bj

Theorem 2.12. Let −1 < λ1 ≤λ2 and 0 ≤μ < 1. Then C μ λ1 αj, Aj  1,q; βj, Bj  1,p  ⊇C μ λ2 αj, Aj  1,q; βj, Bj

Theorem 3.1. Let qz / 0 be univalent in U such that zq′z/qz is starlike univalent in U and R  1  α γ qz  zq′′z q′z −zq′z qz  > 0, α, γ ∈C, γ / 0. 3.1 If f ∈A satisfies the subordination α zIλ αj, Aj

Corollary 3.2. Let the assumptions of Theorem 2.1 hold. Then the subordination α −γ zIλ αj, Aj  1,q; βj, Bj  1,p fz′ Iλ αj, Aj  1,q; βj, Bj  1,p

Corollary 3.3. If f ∈A and assume that 3.1 holds then 1  zIλ αj, Aj  1,q; βj, Bj  1,p fz′′ Iλ αj, Aj  1,q; βj, Bj  1,p

Corollary 3.4. If f ∈A and assume that 3.1 holds then 1  zIλ αj, Aj  1,q; βj, Bj  1,p fz′′ Iλ αj, Aj  1,q; βj, Bj  1,p

Corollary 3.5. If f ∈A and assume that 3.1 holds then 1  zIλ αj, Aj  1,q; βj, Bj  1,p fz′′ Iλ αj, Aj  1,q; βj, Bj  1,p

Theorem 3.6. Let qz / 0 be convex univalent in the unit disk U. Suppose that R α γ qz  > 0, α, γ ∈C for z ∈U, 3.18 and that zq′z/qz is starlike univalent in U. If zIλαj, Aj1,q; βj, Bj1,pfz′/ΦIλαj, Aj1,q; βj, Bj1,pfz ∈Hq0, 1 ∩Q where f ∈A, α zIλ αj, Aj  1,q; βj, Bj

Theorem 3.7. Let q1z / 0, q2z / 0 be convex univalent in the unit disk U satisfying 3.18 and 3.1, respectively. Suppose that zq′ iz/qiz, i  1, 2, is starlike univalent in U. If zIλ αj, Aj  1,q; βj, Bj  1,p fz′ ΦIλ αj, Aj  1,q; βj, Bj