Results & Lemmas (6)
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Lemma 1. · coeff
Lemma 1. [34] If the function r ∈P is given by the series r(z) = 1 + ℓ1z + ℓ2z2 + ℓ3z3 +..., then the following coefficient estimates hold:…
Lemma 1. [34] If the function r ∈P is given by the series r(z) = 1 + ℓ1z + ℓ2z2 + ℓ3z3 + . . . , then the following coefficient estimates hold: 2ℓ2 = ℓ2 1 + x(4 −ℓ2 1), 4ℓ3 = ℓ3 1 + 2ℓ1(4 −ℓ2 1)x −ℓ1(4 −ℓ2 1)x2 + 2(4 −ℓ2 1)(1 −|x|2)z, for some x, z ∈C with max{|x|, |z|} ≤1.
Theorem 1.
Theorem 1. For µ ∈R+ ∪ 0, let f ∈BΣ(µ; q). Then a2 ≤min |ϑq| µ + q, s 2ϑ2q ψ(q, µ) (21)
Theorem 1. For µ ∈R+ ∪{0}, let f ∈BΣ(µ; q). Then a2 ≤min |ϑq| µ + q, s 2ϑ2q ψ(q, µ) (21)
Theorem 2.
Theorem 2. For η ∈R+ ∪ 0, let f ∈BΣ(µ, q). Then a3 −η a2 2 ≤ |ϑq| µ + q2 + q,
Theorem 2. For η ∈R+ ∪{0}, let f ∈BΣ(µ, q). Then a3 −η a2 2 ≤ |ϑq| µ + q2 + q,
Theorem 3.
Theorem 3. Let f ∈BΣ(µ, q). Then H2,2(f) ≤
Theorem 3. Let f ∈BΣ(µ, q) . Then H2,2(f) ≤
Corollary 1.
Corollary 1. Let f given by (1) be in the class SLΣ(Υ(z); q). Then a2 ≤ ϑq
Corollary 1. Let f given by (1) be in the class SLΣ(Υ(z); q). Then a2 ≤ ϑq
Corollary 2.
Corollary 2. [35] Let f given by (1) be in the class SLΣ(Υ(z)). Then a2 ≤ ϑ
Corollary 2. [35] Let f given by (1) be in the class SLΣ(Υ(z)). Then a2 ≤ ϑ
Definitions (3)
Def 1.
Definition 1. [10] The q-bracket ⌈κ⌋q is defined as follows: ⌈κ⌋q =
Definition 1. [10] The q-bracket ⌈κ⌋q is defined as follows: ⌈κ⌋q =
Def 2.
Definition 2. [10] The q−derivative, also known as the q−difference operator, of a function f is defined by ðq⟨f(z)⟩=
Definition 2. [10] The q−derivative, also known as the q−difference operator, of a function f is defined by ðq⟨f(z)⟩=
Def 3.
Definition 3. A bi-univalent function f of the form (1) belongs to the class BΣ(µ; q) if and only if z1−µ ðq⟨f(z)⟩ f(z) 1−µ ≺Υ(z; q) = 1…
Definition 3. A bi-univalent function f of the form (1) belongs to the class BΣ(µ; q) if and only if z1−µ ðq⟨f(z)⟩ f(z) 1−µ ≺Υ(z; q) = 1 + qϑ2 qz2 1 −ϑq z −qϑ2qz2 , (13)
Function classes studied:
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