Subfamilies of bi-univalent functions governed by Bernoulli polynomials

Theorem 2.1. If a function ϕ ∈σ is a member of the family Tσ(ρ, ν, x), (ρ ⩾0, 0 < ν ⩽1), then |d2| ⩽ s ν2|2x −1|3 |(ν(ν + 1)(1 + 5ρ) + (1 −ν)(1 + 2ρ)2)(2x −1)2 −2(ν + 1)2(1 + 2ρ)2(x2 −x + 1 6)|, (2.1) |d3| ⩽ |ν(2x −1)|2 4(ν + 1)2(1 + 2ρ)2 + |ν(2x −1)| 6(ν + 1)(1 + 3ρ), (2.2) and for ξ ∈R |d3 −ξd2

Corollary 2.2. If a function ϕ ∈σ is a member of the family Tσ(ρ, ν, x), (ρ ⩾0, 0 < ν ⩽1), then |d3 −d2 2| ⩽ |ν(2x−1)| 6(ν+1)(1+3ρ).

Theorem 2.3. If a function ϕ ∈σ is a member of the family Wσ(β, τ, ν, x), (0 ⩽β ⩽1, τ ⩾1, 0 < ν ⩽1), then |d2| ⩽ s ν2|2x −1|3 |(ν(ν + 1)(X + S) + (1 −ν)Y2)(2x −1)2 −(ν + 1)2Y2(x2 −x + 1 6)|, (2.20) |d3| ⩽ν2|2x −1|2 (ν + 1)2Y2 + ν|2x −1| (ν + 1)X, (2.21) and for ξ ∈R, |d3 −ξd2 2| ⩽ 

Corollary 3.2. If a function ϕ ∈σ is a member of the family Cσ(ν, x), (0 < ν ⩽1), then |d2| ⩽ν|(2x −1)| s |2x −1| |(ν2 + 1)(2x −1)2 −2(ν + 1)2(x2 −x + 1 6)|, |d3| ⩽ν2(2x −1)2 4(ν + 1)2 + ν|(2x −1)| 6(ν + 1) , and for ξ ∈R, |d3 −ξd2 2| ⩽  

Corollary 3.3. If a function ϕ ∈σ is a member of the family Cσ(ν, x), then |d3 −d2 2| ⩽ν|2x−1| 6(ν+1) . Example 3.4. Letting ν = 1 in the family Tσ(ρ, ν, x), we get a subfamily Fσ(ρ, x) ≡Tσ(ρ, 1, x) of functions ϕ ∈σ satisfying (σϕ′(σ) + ρσ2ϕ′′(σ))′ ϕ′(σ) ≺B(x, σ), (wψ′(w) + ρw2ψ′′(w))′ ψ′(w) ≺B(x, w), where ρ ⩾0, B(x, σ) is as in (1.3), ψ(w) = ϕ−1(w) is as in (1.2), and σ, w ∈U.

Corollary 3.5. If a function ϕ ∈σ is a member of the family Fσ(ρ, x), then |d2| ⩽|2x −1| s |2x −1| 2|(1 + 5ρ)(2x −1)2 −4(1 + 2ρ)2(x2 −x + 1 6)|, |d3| ⩽ (2x −1)2 16(1 + 2ρ)2 + |2x −1| 12(1 + 3ρ), and |d3 −ξd2 2| ⩽ 

Corollary 3.7. For any function ϕ ∈Gσ(τ, ν, x), the upper bounds of |d2|, |d3|, and |d3 −ξd2 2|, ξ ∈R, are given by (2.20), (2.21), and (2.22), respectively, with X = X1 = 3τ, Y = Y1 = 2τ, and S = S1 = 2τ(τ −1). X1, Y1, and S1 are to be used in place of X, Y, and S for Υ in (2.23). Example 3.8. Letting β = 1 in Wσ(β, τ, ν, x), we get a subclass Hσ(τ, ν, x) ≡Wσ(1, τ, ν, x) of functions ϕ ∈σ satisfying 1 2 σ(ϕ′(σ))τ ϕ(σ)  + σ(ϕ′(σ))τ ϕ(σ)  1

Corollary 3.9. For any function ϕ ∈Hσ(τ, ν, x), the upper bounds of |d2|, |d3|, and |d3 −ξd2 2|, ξ ∈R, are given by (2.20), (2.21), and (2.22), respectively, with X = X2 = 3τ −1, Y = Y2 = 2τ −1, and S = S2 = 2τ2 −4τ + 1. X2, Y2, and S2 are to be used in place of X, Y, and S for Υ in (2.23). Example 3.10. Letting ν = 1 in Wσ(β, τ, ν, x), we get a family Kσ(β, τ, x) ≡Wσ(β, τ, 1, x) of functions ϕ ∈σ satisfying σ(ϕ′(σ))τ (1 −β)σ + βϕ(σ) ≺B(x, σ), w(ψ′(w))τ (1 −β)w + βψ(w) ≺B(x, w), where τ ⩾1, 0 ⩽β

Corollary 3.11. If a function ϕ ∈σ is a member of the family Kσ(β, τ, x), (0 ⩽β ⩽1, τ ⩾1), then |d2| ⩽ s |2x −1|3 2|(X + S)(2x −1)2 −2Y2(x2 −x + 1 6)|, |d3| ⩽|2x −1|2 4Y2 + |2x −1| 2X , and for ξ ∈R, |d3 −ξd2 2| ⩽ 

Corollary 3.13. For any function ϕ ∈Mσ(β, ν, x), the upper bounds of |d2|, |d3|, and |d3 −ξd2 2|, ξ ∈R, are given by (2.20), (2.21), and (2.22), respectively, with X = X3 = 3 −β, Y = Y3 = 2 −β, and S = S3 = β2 −β). X3, Y3, and S3 are to be used in place of X, Y, and S for Υ in (2.23).

Definition 1.1. Let ρ ⩾0 and 0 < ν ⩽1. If ϕ ∈σ satisfies 1 2   (σϕ′(σ) + ρσ2ϕ′′(σ))′ ϕ′(σ)  + (σϕ′(σ) + ρσ2ϕ′′(σ))′

Definition 1.2. Let 0 ⩽β ⩽1, τ ⩾1, and 0 < ν ⩽1. If ϕ ∈σ satisfies 1 2  σ(ϕ′(σ))τ (1 −β)σ + βϕ(σ)  +  σ(ϕ′(σ))τ