Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function

Lemma 2.1 Let the function p be of the form (1.2), then |cn| ≤2, n ≥1, (2.1) |cn+k – μcnck| < 2, for 0 ≤μ ≤1 (2.2) |cmcn – ckcl| ≤4 for n + m = k + l, (2.3) cn+2k – μcnc2 k  ≤2(1 + 2μ) for μ ∈R. (2.4)

Lemma 2.2 [47]. Let the function p ∈P be given by (1.2)), then c3 – 2Bc1c2 + Dc3 1  ≤2, if 0 ≤B ≤1, and B(2B – 1) ≤D ≤B.

Lemma 2.3 [48, 49]. Let the function p ∈P be given by (1.2)), then 2c2 = c2 1 + x  4 – c2 1

Lemma 2.4 [50]. Consider the function p ∈P of the form (1.2), 0 < γ < 1, 0 < α < 1, and 8γ (1 – γ )  (αβ – 2λ)2 +  α(γ + α) – β 2 + α(1 – α)(β – 2γ α)2 ≤4α2γ (1 – α)2(1 – γ ). (2.7) Then λb4 1 + γ b2 2 + 2αb1b3 – 3 2βb2

Theorem 3.1 If χℑ∈Rm sin(ℑ) where χℑhas the form (1.5), then |d2| ≤ 1 2(2mℑ(v)), (3.1) |d3| ≤ 1 3(3mℑ(v)), (3.2) |d4| ≤ 1 4(4mℑ(v)), (3.3) |d5| ≤

Theorem 3.3 If a function χℑgiven in (1.5) belongs to the class Rsin(ℑ), then |d2| ≤ 1 2ℑ(v), |d3| ≤ 1 3ℑ(v), |d4| ≤ 1 4ℑ(v), |d5| ≤ 1 5ℑ(v), where ℑ(v) is given by (1.6). The results are sharp for functions (3.13) to (3.16).

Theorem 3.5 Let χℑ∈Rm sin(ℑ). Then, for a complex number ρ, d3 – ρd2 2  ≤ 1 3(3mℑ(v)) max  1,  3ρ(3mℑ(v)) 4(2mℑ(v))2   ,

Corollary 3.6 If χℑ∈Rm sin(ℑ), then for a complex number |ρ| ≤4(2mℑ(v))2 3(3mℑ(v)) , we have d3 – ρd2 2  ≤ 1 3(3mℑ(v)), (3.19) where ℑ(v) is given by (1.6). This inequality is sharp.

Corollary 3.7 [44]. Let χ ∈Rsin. Then, for a complex number ρ, d3 – ρd2 2  ≤1 3 max  1,  3ρ 4   .

Theorem 3.8 Let χℑ∈Rsin(ℑ). Then, for a complex number ρ, d3 – ρd2 2  ≤ 1 3ℑ(v) max  1, ρ(3ℑ(v)) (2ℑ(v))2  , where ℑ(v) is given by (1.6). The result is sharp.

Theorem 3.9 Let χℑ∈Rm sin(ℑ). Then d3 – d2 2  ≤ 1 3(3mℑ(v)), (3.20) where ℑ(v) is given by (1.6). This inequality is sharp for the function χ3(z) = z + 1 3(3mℑ(v))z3 + ··· , z ∈E.

Theorem 3.10 Let χℑ∈Rsin(ℑ). Then d3 – d2 2  ≤ 1 3ℑ(v), where ℑ(v) is given by (1.6). This inequality is sharp for χ3(z) = z + 1 3ℑ(v)z3 + ··· , z ∈E.

Theorem 3.11 Let χℑ∈Rm sin(ℑ). Then |d2d3 – d4| ≤ 1 4(4mℑ(v)). (3.21) This inequality is sharp for χ4,ℑ(z) = z + 1 4(4mℑ(v))z4 + ··· , z ∈E, where ℑ(v) is given by (1.6).

Corollary 3.12 [44]. Let χ ∈Rsin (m = 0 and v = 0). Then |d2d3 – d4| ≤1 4. This inequality is sharp for χ4(z) = z + 1 4z4 + ··· .

Theorem 3.13 Let χℑ∈Rsin(ℑ). Then |d2d3 – d4| ≤ 1 4ℑ(v), where ℑ(v) is given by (1.6). This inequality is sharp for χ4,ℑ(z) = z + 1 4ℑ(v)z4 + ··· , z ∈E.

Theorem 3.15 Let χℑ∈Rm sin(ℑ). Then d2d4 – d2 3  ≤ 1 9(3mℑ(v))2 . (3.22) This inequality is sharp for χ3,ℑ(z) = z + 1 3(3mℑ(v))z3 + ··· , z ∈E, where ℑ(v) is given by (1.6).

Theorem 3.16 Let χℑ∈Rsin(ℑ). Then d2d4 – d2 3  ≤ 1 9(ℑ(v))2 , where ℑ(v) is given by (1.6). This inequality is sharp for χ3,ℑ(z) = z + 1 3ℑ(v)z3 + ··· , z ∈E.

Theorem 3.16. □

Theorem 3.17 Let χℑ∈Rm sin(ℑ). Then H3,1(χ)  ≤ 1 27(3mℑ(v))(3mℑ(v))2 + 1 16(4mℑ(v))(4mℑ(v))

Theorem 3.18 Let χℑ∈Rsin(ℑ). Then H3,1(χ)  ≤ 1 27(ℑ(v))3 + 1 16(ℑ(v))2 + 1 15(ℑ(v))2 , where ℑ(v) is given by (1.6).

Theorem 3.18. □

Theorem 4.1 Let χϕ ∈Rm sin(ℑ). Then H4,1(χ)  ≤ 1 7(7mℑ(v))

Theorem 4.2 Let χϕ ∈Rsin(ℑ). Then H4,1(χ)  ≤ 1 7ℑ(v)