Differential sandwich results for Wanas operator of analytic functions

Lemma 1 ([8]). Let q be univalent in the unit disk U and let θ and φ be analytic in a domain D containing q (U) with φ (w) ̸= 0 when w ∈q (U). Set Q (z) = zq′ (z) φ (q (z)) and h (z) = θ (q (z)) + Q (z). Suppose that (1) Q (z) is starlike univalent in U, (2) ℜ  zh′(z) Q(z)  > 0 for z ∈U. If ξ is analytic in U, with ξ (0) = q (0), ξ (U) ⊂D and (5) θ (ξ (z)) + zξ′ (z) φ (ξ (z)) ≺θ (q (z)) + zq′ (z) φ (q (z)) , then ξ ≺q and q is the best dominant of (5).

Lemma 2 ([9]). Let q be a convex univalent function in Uand let µ ∈C, ν ∈C\{0} with ℜ  1 + zq′′ (z) q′ (z)  > max n 0, −Re µ ν o .

Lemma 3 ([9]). Let q be convex univalent in U and let ν ∈C. Further assume that ℜ(ν) > 0. If ξ ∈H [q (0) , 1]∩Q and ξ (z)+νzξ′ (z) is univalent in U, then (7) q (z) + νzq′ (z) ≺ξ (z) + νzξ′ (z) , which implies that q ≺ξ and q is the best subordinant of (7).

Lemma 4 ([5]). Let q be convex univalent in the unit disk U and let θ and φ be analytic in a domain D containing q (U). Suppose that (1) ℜ  θ′(q(z)) φ(q(z))  > 0 for z ∈U, (2) Q (z) = zq′ (zφ (q (z))) is starlike univalent in U. If ξ ∈H [q (0) , 1]∩Q, with ξ (U) ⊂D, φ (ξ (z))+zξ′ (z) φ (ξ (z)) is univalent in U and (8) θ (q (z)) + zq′ (z) φ (q (z)) ≺θ (ξ (z)) + zξ′ (z) φ (ξ (z)) , then q ≺ξ and q is the best subordinant of (8). 2. Main Results

Theorem 1. Let q be convex univalent in U with q (0) = 1, σ ∈C\{0}, γ > 0 and suppose that q satisfies (9) ℜ  1 + zq′′ (z) q′ (z)  > max n 0, −ℜ γ σ o .

Theorem 2. Let η, τ ∈C, γ > 0, λ ∈C\{0} and q be convex univalent in U with q (0) = 1, q (z) ̸= 0 (z ∈U) and assume that q satisfies (13) ℜ  1 + τ λq (z) + zq′′ (z) q′ (z) −zq′ (z) q (z)  > 0. Suppose that zq′(z) q(z) is starlike univalent in U. If f ∈A satisfies (14) Ω(η, τ, γ, λ, k, δ, α, β; z) ≺η + τq (z) + λzq′ (z)

Theorem 3. Let q be convex univalent in U with q (0) = 1, γ > 0 and ℜ(σ) > 0. Let f ∈A satisfies

Theorem 4. Let η ∈C, γ > 0, λ ∈C\{0} and q be convex univalent in U with q (0) = 1, q (z) ̸= 0 (z ∈U) and assume that q satisfies (22) ℜ τ λq (z)  > 0. Suppose that zq′(z) q(z) is starlike univalent in U. If f ∈A satisfies

Theorem 5. Let q1 and q2 be convex univalent in U with q1 (0) = q2 (0) = 1. Suppose q2 satisfies (9), γ > 0 and ℜ(σ) > 0. Let f ∈A satisfies

Theorem 6. Let q1 and q2 be convex univalent in U with q1 (0) = q2 (0) = 1. Suppose q1 satisfies (22) and q2 satisfies (13). Let f ∈A satisfies

Theorem 3.6, Theorem 3.9], (3) Setting α = 0 and k = β = 1 in Theorems 1, 3 and 5, we get the results obtained by Răducanu and Nechita [11, Corollary 3.3,

Corollary 3.8, Corollary 3.11]. References [1] J. W. Alexander, Functions which map the interior of the unit circle upon simple region, Annals of Mathematics, 17 (1) (1915), 12–22. [2] F. M. Al-Oboudi, On univalent functions defined by a generalized Sălăgean opera- tor, International Journal of Mathematics and Mathematical Sciences, 27 (2004), 1429–1436. [3] A. A. Attiya, M. F. Yassen, Some subordination and superordination results associated with generalized Srivastava-Attiya operator, Filomat,