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Abstract
In this paper, we define certain subclasses of analytic and univalent func-
tions associated with quasi-subordination and we derive the bounds for the
Fekete-Szeg¨o functional
a3 −va2
2
for functions belonging to these subclasses.
2000 Mathematics Subject Classification: 30C45.
Key words: analytic and univalent functions, subordination, quasi-subordination,
majorization, Fekete-Szeg¨o inequality.
1
Results & Lemmas (12)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.
Lemma 1. ([8], p.10) If w ∈Ω, then for any complex number v: |w1| ≤1, w2 −vw2 1 ≤1 + (|v| −1) w2 1 ≤max 1, |v|. The result is sharp for the…
Lemma 1. ([8], p.10) If w ∈Ω, then for any complex number v: |w1| ≤1, w2 −vw2 1 ≤1 + (|v| −1) w2 1 ≤max {1, |v|} . The result is sharp for the functions w(z) = z or w(z) = z2. 2 Main results In this section, we shall obtain Fekete-Szeg¨o inequality for functions in the class Kq (λ, β, b, h) .
Theorem 1.
Theorem 1. Let 0 ≤β ≤λ ≤1 and b ∈C 0. If f ∈A of the from (1) belong to the class Kq (λ, β, b, h), then |a2| ≤ |b| B1 2λβ + λ −β + 1 (6)…
Theorem 1. Let 0 ≤β ≤λ ≤1 and b ∈C \ {0}. If f ∈A of the from (1) belong to the class Kq (λ, β, b, h), then |a2| ≤ |b| B1 2λβ + λ −β + 1 (6) and for any v ∈C a3 −va2 2 ≤ |b| B1 2(6λβ + 2λ −2β + 1) max 1,
Theorem 2.1
Theorem 2.1]. ii) For λ = 1, β = 0 and b = 1 in Theorem 1, we obtain result of [15, Theorem 2.4]. For β = 0, the Theorem 1 reduces to…
Theorem 2.1]. ii) For λ = 1, β = 0 and b = 1 in Theorem 1, we obtain result of [15, Theorem 2.4]. For β = 0, the Theorem 1 reduces to following corollary:
Corollary 1.
Corollary 1. Let 0 ≤λ ≤1 and b ∈C 0. If f ∈A of the from (1) belong to the class Kq (λ, 0, b, h), then |a2| ≤|b| B1 λ + 1, and for some v…
Corollary 1. Let 0 ≤λ ≤1 and b ∈C \ {0}. If f ∈A of the from (1) belong to the class Kq (λ, 0, b, h), then |a2| ≤|b| B1 λ + 1, and for some v ∈C a3 −va2 2 ≤ |b| B1 2(1 + 2λ) max 1,
Corollary 2.
Corollary 2. If f ∈A of the from (1) belong to the class Kq (1, 0, 1, h), then |a2| ≤B1 2, and for some v ∈C a3 −va2 2 ≤B1 6 max 1,
Corollary 2. If f ∈A of the from (1) belong to the class Kq (1, 0, 1, h), then |a2| ≤B1 2 , and for some v ∈C a3 −va2 2 ≤B1 6 max 1,
Theorem 2.
Theorem 2. Let 0 ≤β ≤λ ≤1 and b ∈C 0. If f ∈A of the form (1) satisfies 1 b λβz3f′′′(z) + (2λβ + λ −β)z2f′′(z) + zf′(z) λβz2f′′(z) + (λ…
Theorem 2. Let 0 ≤β ≤λ ≤1 and b ∈C \ {0} . If f ∈A of the form (1) satisfies 1 b λβz3f′′′(z) + (2λβ + λ −β)z2f′′(z) + zf′(z) λβz2f′′(z) + (λ −β)zf′(z) + (1 −λ + β)f(z) −1 ≪h(z) −1 (z ∈U), (19) then |a2| ≤ |b| B1 2λβ + λ −β + 1 and for any v ∈C a3 −va2
Theorem 2.3
Theorem 2.3]. ii) For λ = 1, β = 0 and b = 1 in Theorem 2, we obtain result of [15, Theorem 2.5].
Theorem 2.3]. ii) For λ = 1, β = 0 and b = 1 in Theorem 2, we obtain result of [15, Theorem 2.5].
Theorem 3.
Theorem 3. Let 0 ≤β ≤λ ≤1 and b ∈C 0. If f ∈A of the form (1) belong to the class K(λ, β, b, h), then |a2| ≤ |b| B1 2λβ + λ −β + 1 and for…
Theorem 3. Let 0 ≤β ≤λ ≤1 and b ∈C \ {0} . If f ∈A of the form (1) belong to the class K(λ, β, b, h), then |a2| ≤ |b| B1 2λβ + λ −β + 1 and for any v ∈C a3 −va2 2 ≤ |b| B1 2(6λβ + 2λ −2β + 1) max 1,
Lemma 1
Repeated OCR label — verify against source
Lemma 1, we obtain the desired assertion. The results are sharp for the function f given by 1 + 1 b λβz3f′′′(z) + (2λβ + λ −β)z2f′′(z) +…
Lemma 1, we obtain the desired assertion. The results are sharp for the function f given by 1 + 1 b λβz3f′′′(z) + (2λβ + λ −β)z2f′′(z) + zf′(z) λβz2f′′(z) + (λ −β)zf′(z) + (1 −λ + β)f(z) −1 = h(z), 1 + 1 b λβz3f′′′(z) + (2λβ + λ −β)z2f′′(z) + zf′(z) λβz2f′′(z) + (λ −β)zf′(z) + (1 −λ + β)f(z) −1 =
Theorem 4.
Theorem 4. Let 0 ≤β ≤λ ≤1. If f ∈A of the form (1) belong to the class Kq(λ, β, b, h), then for real v and b, we have a3 −va2 2
Theorem 4. Let 0 ≤β ≤λ ≤1. If f ∈A of the form (1) belong to the class Kq(λ, β, b, h), then for real v and b, we have a3 −va2 2
Theorem 5.
Theorem 5. Let 0 ≤β ≤λ ≤1. If f ∈A of the form (1) belong to the class Kq(λ, β, b, h), then for real v and b, we arrive a3 −va2 2 + (v −σ1)…
Theorem 5. Let 0 ≤β ≤λ ≤1. If f ∈A of the form (1) belong to the class Kq(λ, β, b, h), then for real v and b, we arrive a3 −va2 2 + (v −σ1) |a2|2 ≤ |b| B1 2(6λβ + 2λ −2β + 1) (σ1 ≤v ≤σ1 + ρ) (25) and a3 −va2 2 + (σ1 + 2ρ −v) |a2|2 ≤ |b| B1 2(6λβ + 2λ −2β + 1)
Lemma 1
Repeated OCR label — verify against source
Lemma 1, we have a3 −va2 2 + (σ1 + 2ρ −v) |a2|2 ≤ |b| B1 2(6λβ + 2λ −2β + 1) |w2| + 2 |b| B1(6λβ + 2λ −2β + 1) (2λβ + λ −β + 1)2 (v −σ1…
Lemma 1, we have a3 −va2 2 + (σ1 + 2ρ −v) |a2|2 ≤ |b| B1 2(6λβ + 2λ −2β + 1) |w2| + 2 |b| B1(6λβ + 2λ −2β + 1) (2λβ + λ −β + 1)2 (v −σ1 −ρ) |w1|2 +2 |b| B1(6λβ + 2λ −2β + 1) (2λβ + λ −β + 1)2 (σ1 + 2ρ −v) |w1|2
Function classes studied:
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