Abstract
Using the Al-Oboudi type operator, we present and investigate two special families of bi-univalent functions connected
with the activation function φ(s) = 2/(1 + e−s), s ∈R and k-Fibonacci numbers. We derive the bounds on initial coefficients
and the Fekete-Szeg¨o functional for functions of the type gφ(z) = z +
∞
P
j=2
φ(s)djzj in these introduced families. Furthermore,
we present interesting observations of the results investigated.
Results & Lemmas (10)
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Lemma 1.8
Lemma 1.8 ([35]). If the function p ∈P, then |pi| ⩽2 for each i, where P is the set of regular functions p in D, normalized by p(z) = 1 +…
Lemma 1.8 ([35]). If the function p ∈P, then |pi| ⩽2 for each i, where P is the set of regular functions p in D, normalized by p(z) = 1 + p1z + p2z2 + · · · , such that ℜ(p(z)) > 0, z ∈D. 2. Estimates for the family SRSΣ(µ, γ, β, n, φ(s), epk) We begin by obtaining the first two coefficients and the Fekete-Szeg¨o bounds for functions in SRSΣ(µ, γ, β, n, φ(s), epk).
Theorem 2.1.
Theorem 2.1. If the function g ∈SRSΣ(µ, γ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| (1 + β)nφ(s) p |(γ2 −γ(2µ + 3) + 3(2µ + 1))k2tk + (2(µ…
Theorem 2.1. If the function g ∈SRSΣ(µ, γ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| (1 + β)nφ(s) p |(γ2 −γ(2µ + 3) + 3(2µ + 1))k2tk + (2(µ + 1) −γ)2Tk| , (2.1) |d3| ⩽ 1 (1 + 2β)nφ(s) k|tk|
Corollary 2.2.
Corollary 2.2. If the function g ∈SRKΣ(γ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| (1 + β)nφ(s) p |(γ2 −3γ + 3)k2tk + (2 −γ)2Tk|,
Corollary 2.2. If the function g ∈SRKΣ(γ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| (1 + β)nφ(s) p |(γ2 −3γ + 3)k2tk + (2 −γ)2Tk| ,
Corollary 2.4.
Corollary 2.4. If the function g ∈SRLΣ(µ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| (1 + β)nφ(s) p |3(2µ + 1)k2tk + 4(µ + 1)2Tk|, |d3| ⩽ 1…
Corollary 2.4. If the function g ∈SRLΣ(µ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| (1 + β)nφ(s) p |3(2µ + 1)k2tk + 4(µ + 1)2Tk| , |d3| ⩽ 1 (1 + 2β)nφ(s) k|tk| 3(2µ + 1) +
Corollary 2.5.
Corollary 2.5. If the function g ∈SRMΣ(µ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| φ(s)(1 + β)np |(4µ + 1)k2tk + (2µ + 1)2Tk|, |d3| ⩽ 1…
Corollary 2.5. If the function g ∈SRMΣ(µ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| φ(s)(1 + β)np |(4µ + 1)k2tk + (2µ + 1)2Tk| , |d3| ⩽ 1 φ(s)(1 + 2β)n k|tk| 2(3µ + 1) + k3t2
Corollary 2.7.
Corollary 2.7. If the function g ∈SRNΣ(γ, µ, epk), then |d2| ⩽ k √ k |tk| p |(γ2 −γ(2µ + 3) + 3(2µ + 1))k2tk + (2(µ + 1) −γ)2Tk|, |d3| ⩽ …
Corollary 2.7. If the function g ∈SRNΣ(γ, µ, epk), then |d2| ⩽ k √ k |tk| p |(γ2 −γ(2µ + 3) + 3(2µ + 1))k2tk + (2(µ + 1) −γ)2Tk| , |d3| ⩽ k|tk| 3(2µ + 1) −γ + k3t2 k |(γ2 −γ(2µ + 3) + 3(2µ + 1))k2tk + (2(µ + 1) −γ)2Tk|
Theorem 3.1.
Theorem 3.1. Let 0 ⩽γ ⩽1, τ ⩾1, β ⩾0, and n ∈N0. If g ∈SRBΣ(γ, τ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| φ(s)(1 + β)np |(γ2 + (2τ + 1)(τ…
Theorem 3.1. Let 0 ⩽γ ⩽1, τ ⩾1, β ⩾0, and n ∈N0. If g ∈SRBΣ(γ, τ, β, n, φ(s), epk), then |d2| ⩽ k √ k |tk| φ(s)(1 + β)np |(γ2 + (2τ + 1)(τ −γ))k2 tk + (2τ −γ)2Tk| , (3.1) |d3| ⩽ 1 φ(s)(1 + 2β)n k |tk| (3τ −γ) +
Corollary 3.2.
Corollary 3.2. If the function g ∈SRPΣ(τ, β, n, φ(s), epk), then |d2| ⩽ k √ k|tk| φ(s)(1 + β)np |(2τ + 1)τ k2 tk + 4τ2Tk|, |d3| ⩽ 1 φ(s)(1…
Corollary 3.2. If the function g ∈SRPΣ(τ, β, n, φ(s), epk) , then |d2| ⩽ k √ k|tk| φ(s)(1 + β)np |(2τ + 1)τ k2 tk + 4τ2Tk| , |d3| ⩽ 1 φ(s)(1 + 2β)n k|tk| 3τ + k3t2 k
Corollary 3.3.
Corollary 3.3. If the function g ∈SRNΣ(τ, β, n, φ(s), epk), then |d2| ⩽ k √ k|tk| φ(s)(1 + β)np |τ(2τ −1)k2tk + (2τ −1)2Tk|, |d3| ⩽ 1…
Corollary 3.3. If the function g ∈SRNΣ(τ, β, n, φ(s), epk), then |d2| ⩽ k √ k|tk| φ(s)(1 + β)np |τ(2τ −1)k2tk + (2τ −1)2Tk| , |d3| ⩽ 1 φ(s)(1 + 2β)n k|tk| (3τ −1) + k3t2
Corollary 3.5.
Corollary 3.5. If the function g(z) ∈SRQΣ(γ, τ, epk), then |d2| ⩽ k √ k |tk| p |(γ2 + (2τ + 1)(τ −γ))k2 tk + (2τ −γ)2Tk|, |d3| ⩽ k |tk|…
Corollary 3.5. If the function g(z) ∈SRQΣ(γ, τ, epk), then |d2| ⩽ k √ k |tk| p |(γ2 + (2τ + 1)(τ −γ))k2 tk + (2τ −γ)2Tk| , |d3| ⩽ k |tk| (3τ −γ) + k3 t2 k |(γ2 + (2τ + 1)(τ −γ))k2 tk + (2τ −γ)2Tk|
Function classes studied:
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