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Abstract
In this paper, estimates for second and third MacLaurin coefficients of certain subclasses of bi-univalent functions in the open unit disk defined by convolution are determined, and certain special cases are also indicated. The main result extends and improve a recent one obtained by Srivastava et al.
Results & Lemmas (7)
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Lemma 2.1.
Lemma 2.1. Let the function Φ(z) = 1 + ∞ P n=1 hnzn, z ∈D, such that Φ ∈Pm(β). Then, |hn| ≤m(1 −β), n ≥1.
Lemma 2.1. Let the function Φ(z) = 1 + ∞ P n=1 hnzn, z ∈D, such that Φ ∈Pm(β). Then, |hn| ≤m(1 −β), n ≥1.
Theorem 2.1.
Theorem 2.1. Let f(z) = z + ∞ P n=2 anzn be in the class BRk(m; β), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0, then…
Theorem 2.1. Let f(z) = z + ∞ P n=2 anzn be in the class BRk(m; β), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0, then |a2| ≤min (s m(1 −β) |k3| ; m(1 −β)
Corollary 2.1.
Corollary 2.1. Let f(z) = z + ∞ P n=2 anzn be in the class BRk(m; 0), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0, then…
Corollary 2.1. Let f(z) = z + ∞ P n=2 anzn be in the class BRk(m; 0), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0, then |a2| ≤min r m |k3|; m |k2|
Corollary 2.2.
Corollary 2.2. If f(z) = z + ∞ P n=2 anzn is in the class B(β), then |a2| ≤ q 2(1−β) 3, if 0 ≤β ≤1
Corollary 2.2. If f(z) = z + ∞ P n=2 anzn is in the class B(β), then |a2| ≤ q 2(1−β) 3 , if 0 ≤β ≤1
Theorem 2.2.
Theorem 2.2. Let f(z) = z + ∞ P n=2 anzn be in the class BVk(m; α, β), with α ∈C −1, where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If…
Theorem 2.2. Let f(z) = z + ∞ P n=2 anzn be in the class BVk(m; α, β), with α ∈C \ {−1}, where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0 and 2(1 + 2α)k3 −(1 + 3α)k2 2 ̸= 0, then |a2| ≤min (s
Corollary 2.3.
Corollary 2.3. Let f(z) = z + ∞ P n=2 anzn be in the class Sk m(β), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0 and 2k3…
Corollary 2.3. Let f(z) = z + ∞ P n=2 anzn be in the class Sk m(β), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0 and 2k3 −k2 2 ̸= 0, then |a2| ≤min
Corollary 2.4.
Corollary 2.4. Let f(z) = z + ∞ P n=2 anzn be in the class Ck m(β), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0 and 3k3…
Corollary 2.4. Let f(z) = z + ∞ P n=2 anzn be in the class Ck m(β), where k ∈σ has the form k(z) = z + ∞ P n=2 knzn. If k2, k3 ̸= 0 and 3k3 −2k2 2 ̸= 0, then |a2| ≤min
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