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Abstract
In this paper we establish upper bounds for the second and third
coefficients of holomorphic and bi-univalent functions in a new family which involve
the Bazilevič functions and β-pseudo-starlike functions under a new operator joining
Poisson distribution with Ruscheweyh derivative operator. Also, we discuss Fekete-
Szegö problem of functions in this family.
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.
Theorem 2.1.
Theorem 2.1. Assume that α ≥0, β ≥1, δ ∈N0, 0 ≤λ ≤1 and θ > 0. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; h), with h(z) =…
Theorem 2.1. Assume that α ≥0, β ≥1, δ ∈N0, 0 ≤λ ≤1 and θ > 0. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; h), with h(z) = 1 + e1z + e2z2 + · · · , then (2.1) |a2| ≤ |e1| [(1 −λ)(α + 1) + λ(2β −1)] (δ + 1)θe−θ = |e1| A and (2.2) |a3| ≤min ( max ( e1 B
Theorem 2.1
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Theorem 2.1 becomes the following corollary.
Theorem 2.1 becomes the following corollary.
Corollary 2.1.
Corollary 2.1. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; TM(x),N(x)−1), then |a2| ≤ |M(x)| [(1 −λ)(α + 1) + λ(2β −1)] (δ…
Corollary 2.1. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; TM(x),N(x)−1), then |a2| ≤ |M(x)| [(1 −λ)(α + 1) + λ(2β −1)] (δ + 1)θe−θ
Theorem 2.2.
Theorem 2.2. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; h), then a3 −ηa2 2
Theorem 2.2. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; h), then a3 −ηa2 2
Corollary 2.2.
Corollary 2.2. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; TM(x),N(x)−1), then a3 −ηa2 2 ≤|M(x)| B min ( max ( 1,
Corollary 2.2. If f ∈Σ of the form (1.1) is in the class ΥΣ(α, β, δ, λ, θ; TM(x),N(x)−1), then a3 −ηa2 2 ≤|M(x)| B min ( max ( 1,
Definitions (1)
Def 2.1.
Definition 2.1. Assume that α ≥0, β ≥1, δ ∈N0, 0 ≤λ ≤1, θ > 0 and h is analytic in D, h(0) = 1. The function f ∈Σ is in the family ΥΣ(α, β,…
Definition 2.1. Assume that α ≥0, β ≥1, δ ∈N0, 0 ≤λ ≤1, θ > 0 and h is analytic in D, h(0) = 1. The function f ∈Σ is in the family ΥΣ(α, β, δ, λ, θ; h) if it fulfills the subordinations: (1 −λ) z1−α Jδ θf(z) ′ Jδ
Function classes studied:
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