Results & Lemmas (21)
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Lemma 2.2
Lemma 2.2 [6] Let the function q(z) be univalent in the unit disc U and let θ and ϕ be analytic in a domain D containing q(U) with ϕ(w) ̸=…
Lemma 2.2 [6] Let the function q(z) be univalent in the unit disc 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 (i) Q(z) is starlike univalent in U, (ii) ℜ zh′(z) Q(z) > 0 for z ∈U. If p is analytic with p(0) = q(0), p(U) ⊆D and θ(p(z)) + zp′(z)ϕ(p(z)) ≺θ(q(z)) + zq′(z)ϕ(q(z)), (2.2) then
Lemma 2.3
Lemma 2.3 [16] Let q be a convex univalent function in U and let ψ ∈C, γ ∈C∗= C 0 with ℜ 1 + zq′′(z) q′(z) > max 0, −ℜ ψ γ . If…
Lemma 2.3 [16] Let q be a convex univalent function in U and let ψ ∈C, γ ∈C∗= C\{0} with ℜ 1 + zq′′(z) q′(z) > max 0, −ℜ ψ γ . If p(z) is analytic in U with p(0) = q(0) and ψp(z) + γ zp′(z) ≺ψq(z) + γ zq′(z),
Lemma 2.4
Lemma 2.4 [5] Let q(z) be convex univalent in the unit disc U and let θ and ϕ be analytic in a domain D containing q(U). Suppose that (i) ℜ…
Lemma 2.4 [5] Let q(z) be convex univalent in the unit disc U and let θ and ϕ be analytic in a domain D containing q(U). Suppose that (i) ℜ θ′(q(z)) ϕ(q(z)) > 0 for z ∈U; (ii) zq′(z)ϕ(q(z)) is starlike univalent in U. If p(z) ∈H[q(0), 1] ∩Q, with p(U) ⊆D, and θ(p(z)) + zp′(z)ϕ(p(z)) is univalent in U, and θ(q(z)) + zq′(z)ϕ(q(z)) ≺θ(p(z)) + zp′(z)ϕ(p(z)), (2.4) then q(z) ≺p(z) (z ∈U)
Lemma 2.5
Lemma 2.5 [7] Let q be convex univalent in U and γ ∈C. Further assume that ℜ(γ ) > 0. If p(z) ∈ H[q(0), 1] ∩Q and p(z) + γ zp′(z) is…
Lemma 2.5 [7] Let q be convex univalent in U and γ ∈C. Further assume that ℜ(γ ) > 0. If p(z) ∈ H[q(0), 1] ∩Q and p(z) + γ zp′(z) is univalent in U, then q(z) + γ zq′(z) ≺p(z) + γ zp′(z), (2.5) implies q(z) ≺p(z) (z ∈U) and q is the best subordinant. The last lemma gives us a necessary and sufficient condition for the univalence of a special function which will be used in some particular case.
Lemma 2.6
Lemma 2.6 [11] The function q(z) = (1 −z)−2ab is univalent in the unit disc U if and only if |2ab −1| ≤1 or |2ab + 1| ≤1. 3 Subordination…
Lemma 2.6 [11] The function q(z) = (1 −z)−2ab is univalent in the unit disc U if and only if |2ab −1| ≤1 or |2ab + 1| ≤1. 3 Subordination results Unless otherwise mentioned, we assume throughout this paper that λ > 0, p, n ∈N and the powers understood as principal values.
Theorem 3.1
Theorem 3.1 Let q be univalent in U, with q(0) = 1, and suppose that ℜ 1 + zq′′(z) q′(z) > max 0; −p2 λ ℜ 1 μ (μ ∈C∗; z ∈U).…
Theorem 3.1 Let q be univalent in U, with q(0) = 1, and suppose that ℜ 1 + zq′′(z) q′(z) > max 0; −p2 λ ℜ 1 μ (μ ∈C∗; z ∈U). (3.1)
Corollary 3.2
Corollary 3.2 Let μ ∈C∗and max 0; −p2 λ ℜ 1 μ ≤1 −|B| 1 + |B|. If f ∈A(p) satisfies the following subordination condition μ p I n−1
Corollary 3.2 Let μ ∈C∗and max 0; −p2 λ ℜ 1 μ ≤1 −|B| 1 + |B|. If f ∈A(p) satisfies the following subordination condition μ p I n−1
Theorem 3.4
Theorem 3.4 Let q(z) be univalent in U, with q(0) = 1 and q(z) ̸= 0 for all z ∈U. Let γ, μ ∈C∗and ν, η ∈C∗with ν + η ̸= 0. Let f ∈A(p) and…
Theorem 3.4 Let q(z) be univalent in U, with q(0) = 1 and q(z) ̸= 0 for all z ∈U. Let γ, μ ∈C∗and ν, η ∈C∗with ν + η ̸= 0. Let f ∈A(p) and suppose that f and q satisfy the next conditions: νI n−1 p,λ f (z) + ηI n p,λ f (z) (ν + η)z p ̸= 0 (z ∈U), (3.11) and ℜ 1 + zq′′(z) q′(z) −zq′(z) q(z)
Corollary 3.5
Corollary 3.5 Let μ ∈C∗. Let f ∈A(p) and suppose that I n p,λ f (z) z p ̸= 0 (z ∈U). If 1 + μ z(I n p,λ f (z))′ I n p,λ f (z) −p
Corollary 3.5 Let μ ∈C∗. Let f ∈A(p) and suppose that I n p,λ f (z) z p ̸= 0 (z ∈U). If 1 + μ z(I n p,λ f (z))′ I n p,λ f (z) −p
Corollary 3.6
Corollary 3.6 [8] Let a, b ∈C∗such that |2ab −1| ≤1 or |2ab + 1|leq1. Let f ∈A and suppose that f (z) z ̸= 0 for all z ∈U. If 1 + 1 b zf…
Corollary 3.6 [8] Let a, b ∈C∗such that |2ab −1| ≤1 or |2ab + 1|leq1. Let f ∈A and suppose that f (z) z ̸= 0 for all z ∈U. If 1 + 1 b zf ′(z) f (z) −1 ≺1 + z 1 −z , then f (z) z a
Corollary 3.8
Corollary 3.8 Let −1 ≤A < B ≤1 with B ̸= 0, and suppose that | μ(A−B) B −1| ≤1 or | μ(A−B) B + 1| ≤1. Let f ∈A such that f (z) z ̸= 0 for…
Corollary 3.8 Let −1 ≤A < B ≤1 with B ̸= 0, and suppose that | μ(A−B) B −1| ≤1 or | μ(A−B) B + 1| ≤1. Let f ∈A such that f (z) z ̸= 0 for all z ∈U, and let μ ∈C∗. If 1 + μ zf ′(z) f (z) −1 ≺1 + [B + μ(A −B)]z 1 + Bz then
Theorem 3.4
Theorem 3.4, we obtain the following result due to Aouf et al. [2, Theorem 1]. 123
Theorem 3.4, we obtain the following result due to Aouf et al. [2, Theorem 1]. 123
Corollary 3.9
Corollary 3.9 [2]Leta, b ∈C∗and|τ| < π 2 andsupposethat|2ab cos τe−iτ−1| ≤1or|2ab cos τe−iτ+1| ≤ 1. Let f ∈A and suppose that f (z) z ̸= 0…
Corollary 3.9 [2]Leta, b ∈C∗and|τ| < π 2 andsupposethat|2ab cos τe−iτ−1| ≤1or|2ab cos τe−iτ+1| ≤ 1. Let f ∈A and suppose that f (z) z ̸= 0 for all z ∈U. If 1 + eiτ b cos τ zf ′(z) f (z) −1 ≺1 + z 1 −z then f (z)
Theorem 3.10
Theorem 3.10 Let q be univalent in U, with q(0) = 1, let μ, γ ∈C∗and let δ, , ν, η ∈C with ν + η ̸= 0. Let f (z) ∈A(p) and suppose that f…
Theorem 3.10 Let q be univalent in U, with q(0) = 1, let μ, γ ∈C∗and let δ, , ν, η ∈C with ν + η ̸= 0. Let f (z) ∈A(p) and suppose that f and q satisfy the next two conditions: νI n−1 p,λ f (z) + ηI n p,λ f (z) (ν + η)z p ̸= 0 (z ∈U), (3.19) and ℜ 1 + zq′′(z) q′(z) > max
Corollary 3.11
Corollary 3.11 Let δ ∈C with max 0; −ℜ(δ) ≤1 −|B| 1 + |B|. Let f ∈A(p), and suppose that I n−1 p,λ f (z) z p ̸= 0 for all z ∈U,and let μ…
Corollary 3.11 Let δ ∈C with max{0; −ℜ(δ)} ≤1 −|B| 1 + |B|. Let f ∈A(p), and suppose that I n−1 p,λ f (z) z p ̸= 0 for all z ∈U,and let μ ∈C∗. If I n−1 p,λ f (z) z p μ δ + μ
Corollary 3.12
Corollary 3.12 Let f ∈A such that f (z) z ̸= 0 for all z ∈U, and let μ ∈C∗. If f (z) z μ δ + μ zf ′(z) f (z) −1 + ≺δ 1 + z 1 −z…
Corollary 3.12 Let f ∈A such that f (z) z ̸= 0 for all z ∈U, and let μ ∈C∗. If f (z) z μ δ + μ zf ′(z) f (z) −1 + ≺δ 1 + z 1 −z + + 2z (1 −z)2 , (3.25)
Theorem 4.1
Theorem 4.1 Let q be convex in U with q(0) = 1, let μ ∈C∗with ℜ(μ) > 0. Let f ∈A(p) and suppose that I n p,λ f (z) z p ∈H[q(0); 1] ∩Q. If…
Theorem 4.1 Let q be convex in U with q(0) = 1, let μ ∈C∗with ℜ(μ) > 0. Let f ∈A(p) and suppose that I n p,λ f (z) z p ∈H[q(0); 1] ∩Q. If the function μ p I n−1 p,λ f (z) z p + p −μ p
Corollary 4.2
Corollary 4.2 Let q be convex in U with q(0) = 1, let μ ∈C∗with ℜ(μ) > 0. Let f ∈A(p) and suppose that I n p,λ f (z) z p ∈H[q(0), 1] ∩Q. If…
Corollary 4.2 Let q be convex in U with q(0) = 1, let μ ∈C∗with ℜ(μ) > 0. Let f ∈A(p) and suppose that I n p,λ f (z) z p ∈H[q(0), 1] ∩Q. If the function μ p I n−1 p,λ f (z) z p + p −μ p
Theorem 4.3
Theorem 4.3 Let q be convex in U with q(0) = 1, let μ, γ ∈C∗, and let δ, , ν, η ∈C with ν + η ̸= 0 and ℜ( δ γ ) > 0. Let f ∈A(p) and…
Theorem 4.3 Let q be convex in U with q(0) = 1, let μ, γ ∈C∗, and let δ, , ν, η ∈C with ν + η ̸= 0 and ℜ( δ γ ) > 0. Let f ∈A(p) and suppose that f satisfies the next conditions: νI n−1 p,λ f (z) + ηI n p,λ f (z) (ν + η)z p ̸= 0 (z ∈U), and νI n−1 p,λ f (z) + ηI n p,λ f (z) (ν + η)z p μ
Theorem 4.4
Theorem 4.4 Let q1 and q2 be two convex functions in U with q1(0) = q2(0) = 1, let μ ∈C∗with ℜ(μ) > 0. Let f ∈A(p) and suppose that I n p,λ…
Theorem 4.4 Let q1 and q2 be two convex functions in U with q1(0) = q2(0) = 1, let μ ∈C∗with ℜ(μ) > 0. Let f ∈A(p) and suppose that I n p,λ f (z) z p ∈H[q(0), 1] ∩Q. If the function μ p I n−1 p,λ f (z) z p + p −μ p
Theorem 4.5
Theorem 4.5 Let q1 and q2 be two convex functions in U with q1(0) = q2(0) = 1, let μ, γ ∈C∗, and let δ, , ν, η ∈C with ν + η ̸= 0 and ℜ( δ…
Theorem 4.5 Let q1 and q2 be two convex functions in U with q1(0) = q2(0) = 1, let μ, γ ∈C∗, and let δ, , ν, η ∈C with ν + η ̸= 0 and ℜ( δ γ ) > 0. Let f ∈A(p) and suppose that f satisfies the next conditions: νI n−1 p,λ f (z) + ηI n p,λ f (z) (ν + η)z p ̸= 0 (z ∈U), and νI n−1 p,λ f (z) + ηI n p,λ f (z) (ν + η)z p μ