R. M. El-Ashwah; M. K. Aouf; A. Shamandy; E. E. Ali

Lemma 1 [39]. The sequence {bk}∞ k=1 is a subordinating factor sequence if and only if (1.23) Re  1 + 2 ∞ X k=1 bkzk  > 0 (z ∈U).

Lemma 2 [Rogosinski’s Theorem, 27]. Let h(z) = 1 + ∞ P k=1 ckzk be subordinate to H(z) = 1 + ∞ P k=1 Ckzk in U. If H(z) is univalent in U and H(U) is convex, then |ck| ⩽|C1| , k ⩾1. 319

Theorem 1. If a function f(z) of the form (1.1) is in SPn,q,s λ,l (α1, β1; β, γ), then |a2| ⩽A2. 320

Theorem 2. If a function f(z) of the form (1.1) is in SPn,q,s λ,l (α1, β1; β, γ), then (2.4) |ak| ⩽ 1 Φk,n(α1, λ, l) (P1)k−1 (1)k−1 , k ⩾2, where (2.5) P1 = P1(β, γ) = 

Corollary 1. If a function f(z) of the form (1.1) is in UCVn,q,s λ,l (α1, β1; β, γ), then |a2| ⩽B2. 322

Corollary 2. If a function f(z) of the form (1.1) is in UCVn,q,s λ,l (α1, β1; β, γ), then |ak| ⩽ 1 Φk,n(α1, λ, l) (P1)k−1 (1)k , where P1 = P1(β, γ) is given by (2.5). R e m a r k 1. The results of Theorem 2 and Corollary 1 are sharp for k = 2 or β = 0. R e m a r k 2. Putting q = 2, s = 1, α1 = 2, α2 = 1, β1 = 2 −α (α ̸= 2, 3, 4, . . .) and l = 0 in Theorem 2, we obtain the result obtained by Al-Oboudi and Al-Amoudi [3, Theorem 4].

Theorem 3. A function f(z) of the form (1.1) is in SPn,q,s λ,l (α1, β1; β, γ) if (2.9) ∞ X k=2 [k(1 + β) −(β + γ)] Φk,n(α1, λ, l) |ak| ⩽1 −γ, where Φk,n(α1, λ, l) is defined by (1.13). P r o o f. It suffices to show that β

Corollary 3. A function f(z) of the form (1.1) is in UCVn,q,s λ,l (α1, β1; β, γ) if (2.10) ∞ X k=2 k [k(1 + β) −(β + γ)] Φk,n(α1, λ, l) |ak| ⩽1 −γ. R e m a r k 3. Putting q = 2, s = 1, α1 = 2, α2 = 1, β1 = 2 −α (α ̸= 2, 3, 4, . . .) and l = 0 in Theorem 3 and Corollary 3 we obtain the results obtained by Al-Oboudi and Al-Amoudi [3, Theorem 3.3 and Corollary 3.3, respectively]. 3. Subordination results Employing the techniques used earlier by Attiya [7], Srivastava and Attiya [35], and Singh [34]

Theorem 4. Let f(z) ∈A defined by (1.1) be in the class bSPn,q,s λ,l (α1, β1; β, γ). Then [2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) 2 {[2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) + (1 −γ)}(f ∗ϕ)(z) ≺ϕ(z) (3.1) (z ∈U; ϕ ∈CV), and (3.2) Re {f(z)} > −[2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) + (1 −γ) [2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) (z ∈U). The constant [2(1+β)−(β+γ)]Φ2,n(α1,λ,l)

Theorem 2.4]. (ii) Putting q = 2, s = 1, α1 = a (a > 0), α2 = 1, β1 = c (c > 0) and l = 0 in

Theorem 4, we obtain the result obtained by Prajapat and Raina [26 Theorem 1]. (iii) Putting q = 2, s = 1, α1 = a (a > 0), α2 = 1, β1 = c (c > 0) and l = λ = 0 in

Theorem 4, we obtain the result obtained by Frasin [14, Theorem 2.1]. Similarly, by using (1.17) and Theorem 4 we can prove the following corollary.

Corollary 4. Let f(z) ∈A defined by (1.1) be in the class bUCVn,q,s λ,l (α1, β1; β, γ). Then [2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) {2 [2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) + (1 −γ)}(f ∗ϕ)(z) ≺ϕ(z) (3.9) (z ∈U; ϕ ∈CV), and (3.10) Re {f(z)} > −2 [2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) + (1 −γ) 2 [2(1 + β) −(β + γ)] Φ2,n(α1, λ, l) (z ∈U). The constant [2(1+β)−(β+γ)]Φ2,n(α1,λ,l)

Corollary 5. Let f(z) ∈A defined by (1.1) be in the class bSP(m; β, γ). Then (3.11) [2(1 + β) −(β + γ)] Imf(z) 2 {[2(1 + β) −(β + γ)] Imf(z) + (1 −γ)}(f ∗ϕ)(z) ≺ϕ(z) (z ∈U; ϕ ∈CV), and Re {f(z)} > −[2(1 + β) −(β + γ)] Imf(z) + (1 −γ) [2(1 + β) −(β + γ)] Imf(z) (z ∈U). The constant [2(1+β)−(β+γ)]Imf(z) 2{[2(1+β)−(β+γ)]Imf(z)+(1−γ)} is the best estimate. Putting λ = l = 0, n = 1, q = 2, s = 1, α1 = µ (µ > 0), α2 = 1 and β1 = δ + 1 (δ > −1) in Theorem 4, we obtain the following corollary.

Corollary 6. Let f(z) ∈A defined by (1.1) be in the class bSP (µ, δ; β, γ). Then [2(1 + β) −(β + γ)] Iδ,µf(z) 2 {[2(1 + β) −(β + γ)] Iδ,µf(z) + (1 −γ)} (f ∗ϕ)(z) ≺ϕ(z) (3.12) (z ∈U; ϕ ∈CV), and Re {f(z)} > −[2(1 + β) −(β + γ)] Iδ,µf(z) + (1 −γ) [2(1 + β) −(β + γ)] Iδ,µf(z) (z ∈U). The constant [2(1+β)−(β+γ)]Iδ,µf(z) 2{[2(1+β)−(β+γ)]Iδ,µf(z)+(1−γ)} is the best estimate. Putting λ = l = 0, n = 1, q = 2, s = 1, α1 = ν + 1, α2 = 1 and β1 = ν + 2 (ν > −1) in Theorem 4, we obtain the following corollar

Corollary 7. Let f(z) ∈A defined by (1.1) be in the class bSP(ν; β, γ). Then (3.13) [2(1 + β) −(β + γ)] Jνf(z) 2 {[2(1 + β) −(β + γ)] Jνf(z) + (1 −γ)} (f ∗ϕ)(z) ≺ϕ(z) (z ∈U; ϕ ∈CV), and Re {f(z)} > −[2(1 + β) −(β + γ)] Jνf(z) + (1 −γ) [2(1 + β) −(β + γ)] Jνf(z) (z ∈U). The constant [2(1+β)−(β+γ)]Jνf(z) 2{[2(1+β)−(β+γ)]Jνf(z)+(1−γ)} is the best estimate. Putting q = 2, s = 1, α1 = µ (µ > 0), α2 = 1 and β1 = δ + 1 (δ > −1) in

Theorem 4, we obtain the following corollary. 327

Corollary 8. Let f(z) ∈A defined by (1.1) be in the class bSPn λ,l (µ, δ + 1; β, γ). Then [2(1 + β) −(β + γ)][ l+1+λ l+1 µ δ+1]n 2{[2(1 + β) −(β + γ)][ l+1+λ l+1 µ δ+1]n + (1 −γ)}(f ∗ϕ)(z) ≺ϕ(z) (3.14) (z ∈U; ϕ ∈CV), and Re{f(z)} > −

Theorem 4, we obtain the following corollary.

Corollary 9. Let f(z) ∈A defined by (1.1) be in the class bSPn λ,l (δ +1, a, c; β, γ). Then [2(1 + β) −(β + γ)][ l+1+λ l+1 a(δ+1) c ]n 2{[2(1 + β) −(β + γ)][ l+1+λ l+1 a(δ+1) c ]n + (1 −γ)} (f ∗ϕ)(z) ≺ϕ(z) (3.15)

Corollary 10. Let f(z) ∈A defined by (1.1) be in the class bSPn λ,l (2, m+1; β, γ). Then [2(1 + β) −(β + γ)][ l+1+λ l+1 2 m+1]n 2{[2(1 + β) −(β + γ)][ l+1+λ l+1 2 m+1]n + (1 −γ)} (f ∗ϕ)(z) ≺ϕ(z) (3.16) (z ∈U; ϕ ∈CV), 328

Corollary 11. Let f(z) ∈A defined by (1.1) be in the class bSPn λ,l (ν; β, γ). Then [2(1 + β) −(β + γ)][ l+1+λ l+1 ν+1 ν+2]n 2{[2(1 + β) −(β + γ)][ l+1+λ l+1 ν+1 ν+2]n + (1 −γ)} (f ∗ϕ)(z) ≺ϕ(z) (3.17) (z ∈U; ϕ ∈CV), and Re{f(z)} > − [2(1 + β) −(β + γ)][ l+1+λ

Corollary 3]. (ii) Putting γ = β = 0, n = 1, λ = l = 0, q = 2, s = 1 and α1 = α2 = β1 = 1 in

Theorem 4, we obtain the result obtained by Singh [33, Corollary 2.2]. A c k n o w l e d g m e n t s. The authors thank the referees for their valuable sug- gestions improving the paper. References [1] R. Aghalary, Gh. Azadi: The Dziok-Srivastava operator and k-uniformly starlike func- tions. J. Inequal. Pure Appl. Math. 6 (2005), 1–7. [2] F. M. Al-Oboudi: On univalent functions defined by a generalized Salagean operator. Internat. J. Math. Math Sci. 27 (2004), 1429–1436. [3] F. M. Al-Oboudi, K.