Abstract
In this paper, we define a class of analytic functions, F(H, α, δ, µ),
satisfying the following condition
α
zf′(z)
f(z)
δ
+ (1 −α)
zf′(z)
f(z)
µ
1 + zf′′(z)
f′(z)
1−µ!
≺H(z, t),
where α ∈[0, 1], δ ∈[1, 2] and µ ∈[0, 1].
We give coefficient estimates and Fekete-Szeg¨o inequality for this
class.
1
Results & Lemmas (5)
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.
Corollary 1
Corollary 1 [5] Let f ∈A and also let α ∈[0, 1], a ∈[0, 1], δ ∈[1, 2] and µ ∈[0, 1]. If ℜ
Corollary 1 [5] Let f ∈A and also let α ∈[0, 1], a ∈[0, 1], δ ∈[1, 2] and µ ∈[0, 1]. If ℜ
Lemma 1
Lemma 1 [6] Let the Schwarz function w be given by w(z) = w1z + w2z2 + w3z3 + · · ·, z ∈U. (4) Then |w1| ≤1, |w2 −tw2 1| ≤1 + (|t| −1)|w1|2…
Lemma 1 [6] Let the Schwarz function w be given by w(z) = w1z + w2z2 + w3z3 + · · · , z ∈U. (4) Then |w1| ≤1, |w2 −tw2 1| ≤1 + (|t| −1)|w1|2 ≤max{1, |t|}, where t ∈C.
Theorem 1
Theorem 1 Let f ∈A of the form (1) belong to the class F(H, α, δ, µ). Then |a2| ≤ 2t αδ + (1 −α)(2 −µ), (5) and, for λ ∈C, a3 −λa2 2 ≤ t αδ…
Theorem 1 Let f ∈A of the form (1) belong to the class F(H, α, δ, µ). Then |a2| ≤ 2t αδ + (1 −α)(2 −µ), (5) and, for λ ∈C, a3 −λa2 2 ≤ t αδ + (1 −α)(3 −2µ) max 1, 2t
Corollary 2
Corollary 2 [2] Let f ∈A of the form (1) satisfying the condition
Corollary 2 [2] Let f ∈A of the form (1) satisfying the condition
Corollary 3
Corollary 3 [1] Let f ∈A of the form (1) satisfying the condition zf′(z) f(z) µ 1 + zf′′(z) f′(z) 1−µ ≺H(z, t), where µ ∈[0, 1]. Then…
Corollary 3 [1] Let f ∈A of the form (1) satisfying the condition zf′(z) f(z) µ 1 + zf′′(z) f′(z) 1−µ ≺H(z, t), where µ ∈[0, 1]. Then |a2| ≤ 2t 2 −µ, and, for λ ∈C, a3 −λa2 2