Texture enhancement of skin lesion images via Hankel determinants of $\lambda$-generalized Sakaguchi type functions in symmetric domain

Lemma 3.1. [21] Let p ∈P, given in (1.4). Then, |cn| ≤2, ∀n ≥1. (3.1)

Lemma 3.2. [21] Let p ∈P be given by (1.4). Then, |cn+k −θcnck| ≤2 max {1, |2θ −1|} =  2, for 0 ≤θ ≤1; 2|2θ −1|, otherwise. (3.2)

Lemma 3.3. [22,23] Let p ∈P, as given by (1.4). Then, if Q ∈[0, 1] with Q(2Q −1) ≤Q ≤T, we have |c3 −2Qc1c2 + Tc3 1| ≤2. (3.3)

Lemma 3.4. [23–25] Let p ∈P, be in the form (1.4). Then, for x, y ∈D, we have 2c2 = c2 1 + x(4 −c2 1), (3.4) 4c3 = c3 1 + 2c1x(4 −c2 1) −c1x2(4 −c2 1) + 2(4 −c2 1)(1 −|x|2)δ, (3.5) 8c4 =c4 1 + (4 −c2 1)x h

Lemma 3.5. [26] Let α, β, γ, and u satisfy u, α ∈(0, 1) and 8u(1 −u)[(ζη −2γ)2 + (α(u + ζ) −η)2] + ζ(1 −ζ)(η −2uζ)2 ≤4uζ2(1 −ζ)2(1 −u). (3.7) If p ∈P, is in the form (1.4). Then, |γc4 1 + uc2 2 + 2ζc1c3 −3 2ηc2 1c2 −c4| ≤2. (3.8) AIMS Mathematics Volume 11, Issue 6, 16811–16836.

Theorem 4.1. Let f ∈A be given in (1.1). If f ∈S∗(φH), then the coefficients r2,r3,r4, and r5 are |r2| ≤ 1 2(1 −2λ + 2λ2), |r3| ≤ 1 [3(1 −λ)2 + 2λ2], |r4| ≤ 1 4[(1 −λ)2 + λ2] , |r5| ≤ 1 [5(1 −λ)2 + 4λ2]. (4.1)

Theorem 4.2. Let f ∈A be given in (1.1). If f ∈S∗(φH), then |r3 −r2 2| ≤ 1 (3(1 −λ)2 + 2λ2). (4.13) This is the best possible.

Theorem 4.3. Let f ∈A be in the form (1.1). If f ∈S∗(φH), then |r4 −r2r3| ≤ 1 4(1 −2λ + 2λ2). (4.14) This inequality is the best possible.

Theorem 4.4. Let f ∈A be given by (1.1). If f ∈S∗(φH), then |r2r4 −r2 3| ≤ 1 (3(1 −λ)2 + 2λ2)2. (4.15) This inequality is the best possible. AIMS Mathematics Volume 11, Issue 6, 16811–16836.

Theorem 4.5. Let f ∈A as given in (1.1). If f ∈S∗(φH), then |H3,1(f)| ≤ Ω(λ) 48D5α2β2. (4.16)

Corollary 4.1. For, S∗(λ) = f ∈K : (1 −λ)f ′(z)
1−λ + λ

Definition 1.1. The class S∗(λ), known as the linear combination of functions that maps the open unit disk D onto a bounded turning domain and a symmetric starlike domain [4], is defined as S∗(λ) = f ∈K : (1 −λ) f ′(z)
1−λ + λ