Abstract
A bi-univalent function is a univalent function defined on the unit disk with its inverse also univalent on the unit disk. Estimates for the initial coefficients are obtained for bi-univalent functions belonging to certain classes defined by subordination and relevant connections with earlier results are pointed out.
Results & Lemmas (6)
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Theorem 2.2.
Theorem 2.2. If f ∈Rσ(λ,ϕ), then |a2| ≤ r B1 +|B1 −B2| 1+2λ (2.2)
Theorem 2.2. If f ∈Rσ(λ,ϕ), then |a2| ≤ r B1 +|B1 −B2| 1+2λ (2.2)
Theorem 2.5.
Theorem 2.5. If f ∈S ∗σ(ϕ), then |a2| ≤min p B1 +|B2 −B1|, s B2 1 +B1 +|B2 −B1| 2, B1 √B1 q
Theorem 2.5. If f ∈S ∗σ(ϕ), then |a2| ≤min p B1 +|B2 −B1|, s B2 1 +B1 +|B2 −B1| 2 , B1 √B1 q
Theorem 2.8.
Theorem 2.8. If f ∈Kσ(ϕ), then |a2| ≤min s B2 1 +B1 +|B2 −B1| 6, B1 2 and
Theorem 2.8. If f ∈Kσ(ϕ), then |a2| ≤min s B2 1 +B1 +|B2 −B1| 6 , B1 2 and
Theorem 2.10.
Theorem 2.10. Let f ∈σ be given by (1.1). If f ∈K (ϕ) and F ∈R(ϕ), then |a2| ≤ r 3[B1 +|B2 −B1|] 8 and |a3| ≤5[B1 +|B2 −B1|] 12.
Theorem 2.10. Let f ∈σ be given by (1.1). If f ∈K (ϕ) and F ∈R(ϕ), then |a2| ≤ r 3[B1 +|B2 −B1|] 8 and |a3| ≤5[B1 +|B2 −B1|] 12 .
Theorem 2.12.
Theorem 2.12. Let f ∈σ be given by (1.1). If f ∈S ∗(ϕ) and F ∈R(ϕ), then |a2| ≤ p 5[B1 +|B2 −B1|] 3, and |a3| ≤7[B1 +|B2 −B1|] 9.
Theorem 2.12. Let f ∈σ be given by (1.1). If f ∈S ∗(ϕ) and F ∈R(ϕ), then |a2| ≤ p 5[B1 +|B2 −B1|] 3 , and |a3| ≤7[B1 +|B2 −B1|] 9 .
Theorem 2.14.
Theorem 2.14. Let f ∈σ given by (1.1). If f ∈S ∗(ϕ) and F ∈K (ϕ), then |a2| ≤ r B1 +|B2 −B1| 2 and |a3| ≤B1 +|B2 −B1| 2.
Theorem 2.14. Let f ∈σ given by (1.1). If f ∈S ∗(ϕ) and F ∈K (ϕ), then |a2| ≤ r B1 +|B2 −B1| 2 and |a3| ≤B1 +|B2 −B1| 2 .
Function classes studied:
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