Horadam Polynomial coefficient estimates for the classes of $$\lambda $$–bi-pseudo-starlike and Bi-Bazilevič Functions

Coefficient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modified Sigmoid Activation Function and Horadam Polynomials

“…Remark 2.1. For γ = β = 1, Corollary 2.1 reduce to Corollary 2.1 of Magesh et al [25] and Corollary 2.1 further coincide with Corollary 2.1 of Abirami et al [1], when k = 0 and φ(s) = 1. Corollary 2.1 coincide with Theorem 2.2 of Alamoush [3], when γ = k = 0 and φ(s) = 1 and also we obtain Corollary 1 and Corollary 3 of [31] for k = 0, γ = φ(s) = 1.…”

“…1+2β) k φ(s)|(τ (2τ −1))(bx) 2 −(2τ −1) 2 (pbx 2 +qa)| ; 1 − (1+2β) k δ (1+β) 2k φ(s) ≥ Ω 2 ,whereΩ 2 = 1 (3τ −1) (τ (2τ − 1)) − (2τ − 1) 2Remark Corollary 3.2 reduces to Theorem 2.1 of[25], when β = 1 and also the results of Corollary 3.2 coincide with Theorem 2.1 of Abirami et al[1], when k = 0 and φ(s) = 1.Corollary 3.3. If the function g ∈ SQ (x, γ, τ ), then |d 2 | ≤ |bx| |bx| |(γ 2 + (τ − γ)(2τ + 1))(bx) 2 − (2τ − γ) 2 (pbx 2 + qa)| , |d 3 | ≤ (bx) 2 (2τ − γ) 2 + |bx| (3τ − γ)and for δ ∈ R,|d 3 − δd 2 |1 − δ| ≤ Ω 3 |1−δ||bx| 3 |(γ 2 +(τ −γ)(2τ +1))(bx) 2 −(2τ −γ) 2 (pbx 2 +qa)| ; |1 − δ| ≥ Ω 3 ,whereΩ 3 = 1 (3τ − γ) (γ 2 + (τ − γ)(2τ + 1)) − (2τ − γ) 2 pbx 2Remark Corollary 3.3 reduces to Theorem 2.1 of[1], when γ = 1.…”

“…Remark 1.3. i) For µ = γ = 0, the class SN Σ (x, 0, 0) ≡ H Σ (x) was studied by Alamoush [2] and ii) For µ = 0 and γ = 1, the family SN Σ (x, 1, 0) ≡ S * Σ (x) was investigated by Srivastava et al [32]. Remark 1.4. i) For γ = 0, the family SQ Σ (x, 0, τ ) ≡ S * Σ (x, τ ) was investigated by Abirami et al [1] and ii) For β = 1, the family SN (x, τ, k, 1, φ(s)) ≡ M Σ (x, τ, k, φ(s)) was considered in [25].…”

“…Additional information about these polynomials can be found in [7], [8], [16], [17], [24] and [42]. More details about the famous Fekete-Szegö problem connected with Haradam polynomials are available with the works of [1], [2], [3], [26], [31], [38] and [41].…”

Certain classes of bi-univalent functions associated with the Horadam polynomials

“…In 2010, Srivastava et al [28] revived the study of bi-univalent functions by their pioneering work on the study of coefficient problems. Various subclasses of the bi-univalent function class Σ were introduced and non-sharp estimates on the first two coefficients |a 2 | and |a 3 | in the Taylor-Maclaurin series expansion (1) were found in the very recent investigations (see, for example, [1,2,3,4,5,6,7,8,9,10,12,13,16,17,18,19,20,21,22,23,24,25,26,27,29,30]) and including the references therein. The afore-cited all these papers on the subject were actually motivated by the work of Srivastava et al [28].…”