Abstract
In the present paper, we introduce a certain subclass Kq(λ, γ, h)
of analytic functions by means of a quasi-subordination. Sharp bounds of
the Fekete-Szeg˝o functional for functions belonging to the class Kq(λ, γ, h)
are obtained. The results presented in the paper give improved versions
for the certain subclasses involving the quasi-subordination and majoriza-
tion.
1
Results & Lemmas (8)
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 ν: |w1| ≤1, |w2 −νw2 1| ≤1 + (|ν| −1)|w2 1| ≤max 1, |ν|. The result is sharp for…
Lemma 1 ([8], p.10) If w ∈Ω, then for any complex number ν: |w1| ≤1, |w2 −νw2 1| ≤1 + (|ν| −1)|w2 1| ≤max{1, |ν|}. The result is sharp for the functions w(z) = z or w(z) = z2. 2 Main results
Theorem 1
Theorem 1 Let 0 ≤λ ≤1 and γ ∈C 0. If f ∈A of the form (1) belonging to the class Kq(λ, γ, h), then |a2| ≤ |γ|B1 2(2 −λ) (6) and for any ν…
Theorem 1 Let 0 ≤λ ≤1 and γ ∈C\{0}. If f ∈A of the form (1) belonging to the class Kq(λ, γ, h), then |a2| ≤ |γ|B1 2(2 −λ) (6) and for any ν ∈C |a3 −νa2 2| ≤ |γ|B1 3(3 −λ) max
Corollary 1
Corollary 1 If f ∈A of the form (1) satisfies 1 γ(f ′(z) + zf ′′(z) −1) ≺q (h(z) −1) (z ∈U, γ ∈C 0 ), then |a2| ≤|γ|B1 4, and for some ν ∈C…
Corollary 1 If f ∈A of the form (1) satisfies 1 γ(f ′(z) + zf ′′(z) −1) ≺q (h(z) −1) (z ∈U, γ ∈C\{0}), then |a2| ≤|γ|B1 4 , and for some ν ∈C |a3 −νa2 2| ≤|γ|B1 9 max
Theorem 2
Theorem 2 Let 0 ≤λ ≤1 and γ ∈C 0. If f ∈A of the form (1) satisfies 1 γ zf ′(z) + z2f ′′(z) (1 −λ)z + λzf ′(z) −1 ≪(h(z) −1) (z ∈U),…
Theorem 2 Let 0 ≤λ ≤1 and γ ∈C\{0}. If f ∈A of the form (1) satisfies 1 γ zf ′(z) + z2f ′′(z) (1 −λ)z + λzf ′(z) −1 ≪(h(z) −1) (z ∈U), (19) then |a2| ≤ |γ|B1
Theorem 3
Theorem 3 Let 0 ≤λ ≤1 and γ ∈C 0. If f ∈A of the form (1) belonging to the class K(λ, γ, h), then |a2| ≤ |γ|B1 2(2 −λ) and for any ν ∈C |a3…
Theorem 3 Let 0 ≤λ ≤1 and γ ∈C\{0}. If f ∈A of the form (1) belonging to the class K(λ, γ, h), then |a2| ≤ |γ|B1 2(2 −λ) and for any ν ∈C |a3 −νa2 2| ≤ |γ|B1 3(3 −λ) max
Theorem 4
Theorem 4 Let 0 ≤λ ≤1. If f ∈A of the form (1) belonging to the class Kq(λ, γ, h), then for real ν and γ, we have |a3 −νa2 2| ≤ …
Theorem 4 Let 0 ≤λ ≤1. If f ∈A of the form (1) belonging to the class Kq(λ, γ, h), then for real ν and γ, we have |a3 −νa2 2| ≤ |γ|B1 3(3−λ) h B1γ
Theorem 5
Theorem 5 Let 0 ≤λ ≤1. If f ∈A of the form (1) belonging to the class Kq(λ, γ, h), then for real ν and γ, we have |a3 −νa2 2| + (ν…
Theorem 5 Let 0 ≤λ ≤1. If f ∈A of the form (1) belonging to the class Kq(λ, γ, h), then for real ν and γ, we have |a3 −νa2 2| + (ν −σ1)|a2|2 ≤ |γ|B1 3(3 −λ) (σ1 ≤ν ≤σ1 + ρ) (25) and |a3 −νa2 2| + (σ1 + 2ρ −ν)|a2|2 ≤ |γ|B1 3(3 −λ) (σ1 + ρ ≤ν ≤σ1 + 2ρ), (26)
Lemma 1
Lemma 1, we have |a3−νa2 2| + (σ1 + 2ρ −ν)|a2|2 ≤ |γ|B1 3(3 −λ) |w2| + 3|γ|B1(3 −λ) 4(2 −λ)2 (ν−σ1−ρ)|w1|2 + 3|γ|B1(3 −λ) 4(2 −λ)2 (σ1 +…
Lemma 1, we have |a3−νa2 2| + (σ1 + 2ρ −ν)|a2|2 ≤ |γ|B1 3(3 −λ) |w2| + 3|γ|B1(3 −λ) 4(2 −λ)2 (ν−σ1−ρ)|w1|2 + 3|γ|B1(3 −λ) 4(2 −λ)2 (σ1 + 2ρ −ν)|w1|2 ≤ |γ|B1
Function classes studied:
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