Results & Lemmas (19)
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Lemma 1
Lemma 1: Miler and Mocanu (2000). Let q (z) be univalent in the unit disk U and θ and ϕ be analytic in a domain D containing q (U) with (w)…
Lemma 1: Miler and Mocanu (2000). Let q (z) be univalent in the unit disk U and θ and ϕ be analytic in a domain D containing q (U) with (w) 0 ϕ ≠ when w q(U). ∈ Set:
Lemma 2
Lemma 2: Bulboaca (2002). Let q(z) be convex univalent in the unit disk U and ϑ and ϕ be analytic in a domain D containing q(U) Suppose…
Lemma 2: Bulboaca (2002). Let q(z) be convex univalent in the unit disk U and ϑ and ϕ be analytic in a domain D containing q(U) Suppose that : zq (z) (q(z)) ′ φ is starlike univalent in U
Theorem 1
Theorem 1: Let q,q(z)≠ 0 be a univalent function in U and g(z)≠ 0 be analytic in C such that for nonnegative real numbers μ and ν Eq. 2:
Theorem 1: Let q,q(z)≠ 0 be a univalent function in U and g(z)≠ 0 be analytic in C such that for nonnegative real numbers μ and ν Eq. 2:
Theorem 1
Theorem 1 we have
Theorem 1 we have
Corollary 1
Corollary 1: Let q,q(z)≠ 0 be a univalent function in U and zf (z) 0 (z) ′ ≠ Φ be analytic in U satisfy (2). If Eq. 4:
Corollary 1: Let q,q(z)≠ 0 be a univalent function in U and zf (z) 0 (z) ′ ≠ Φ be analytic in U satisfy (2). If Eq. 4:
Theorem 2
Theorem 2 we obtain the following result which can be found in (Singh et al., 2009), Theorem 3.2]:
Theorem 2 we obtain the following result which can be found in (Singh et al., 2009), Theorem 3.2]:
Corollary 2
Corollary 2: Let q,q(z) ≠ 0 be a univalent function in U. If zf (z) 0,z U f (z) ′ ≠ ∈ and Eq. 5:
Corollary 2: Let q,q(z) ≠ 0 be a univalent function in U. If zf (z) 0,z U f (z) ′ ≠ ∈ and Eq. 5:
Theorem 3.3
Theorem 3.3].
Theorem 3.3].
Corollary 3
Corollary 3: Let q,q(z) ≠ 0 be a univalent function in U. If zf (z) 0,z U (f (z)) ′ ≠ ∈ Φ and
Corollary 3: Let q,q(z) ≠ 0 be a univalent function in U. If zf (z) 0,z U (f (z)) ′ ≠ ∈ Φ and
Corollary 4
Corollary 4: Let q,q(z) ≠ 0 be a univalent function in U and z(f (z) (z)) 0 f(z) (z) ′ ∗Φ ≠ ∗Ψ be analytic in U satisfy (2). If the…
Corollary 4: Let q,q(z) ≠ 0 be a univalent function in U and z(f (z) (z)) 0 f(z) (z) ′ ∗Φ ≠ ∗Ψ be analytic in U satisfy (2). If the subordination Eq. 7:
Theorem 2
Theorem 2: Let f∈A and q,q(z) ≠ 0 be a univalent function in U. Assume that z(f (z) (z)) p(z):= 0 f (z) (z) ′ ∗Φ ≠ ∗Ψ is analytic in U…
Theorem 2: Let f∈A and q,q(z) ≠ 0 be a univalent function in U. Assume that z(f (z) (z)) p(z) := 0 f (z) (z) ′ ∗Φ ≠ ∗Ψ is analytic in U satisfies (2-3) for some g If ,
Theorem 3
Theorem 3: Let q(z) be convex univalent in the unit disk U. Suppose that g ia an analytic in the unit disk such that
Theorem 3: Let q(z) be convex univalent in the unit disk U. Suppose that g ia an analytic in the unit disk such that
Theorem 4
Theorem 4: Let 1 2 q (z),q 0 ≠ be convex and univalent in U respectively. Suppose that g ia an analytic in U such that Eq. 8:
Theorem 4: Let 1 2 q (z),q 0 ≠ be convex and univalent in U respectively. Suppose that g ia an analytic in U such that Eq. 8:
Corollary 5
Corollary 5: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A
Corollary 5: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A
Corollary 6
Corollary 6: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A f (z) 0 Hq(0),1] Q,
Corollary 6: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A f (z) 0 Hq(0),1] Q,
Corollary 7
Corollary 7: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A.
Corollary 7: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A.
Corollary 8
Corollary 8: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A
Corollary 8: Let the conditions of Theorem 4 on the functions 1 q and 2 q hold. If for f∈A
Corollary 9
Corollary 9: Let the conditions of Theorem 4 on the functions 1q and 2 q hold. If for f A, ∈
Corollary 9: Let the conditions of Theorem 4 on the functions 1q and 2 q hold. If for f A, ∈
Corollary 10
Corollary 10: Let the assumptions of Theorem 4 on the function:
Corollary 10: Let the assumptions of Theorem 4 on the function:
Definitions (1)
Def 1
Definition 1: (Miller and Mocanu, 2003) Denote by Q the set of all functions f(z) that are analytic and injective on U E(f ) −
Definition 1: (Miller and Mocanu, 2003) Denote by Q the set of all functions f(z) that are analytic and injective on U E(f ) −
Function classes studied:
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