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Abstract
In the current study, we investigate a subclass of convex functions, denoted by
𝒦n–1,ℒ, associated with a domain bounded by an epicycloid with n – 1 cusps. The
primary objective is to derive sharp bounds for the coefficient bounds, the
Fekete–Szegö inequality, and the second Hankel determinant, as well as to establish
an upper bound for the third Hankel determinant for this newly introduced class.
Mathematics Subject Classification: 30C45; 30C50
Results & Lemmas (12)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1
Lemma 1 [22] Assuming that p ∈𝒫with the series expansion of the form (2.1). Then, for x,δ,ρ ∈𝔻, we have 2c2 = c1 2 + (4 – c1 2)x, (2.2) 4c3…
Lemma 1 [22] Assuming that p ∈𝒫with the series expansion of the form (2.1). Then, for x,δ,ρ ∈𝔻, we have 2c2 = c1 2 + (4 – c1 2)x, (2.2) 4c3 = c1 3 + 2c1x(4 – c1 2) – x2c1(4 – c1 2) + 2(1 – |x|2)(4 – c1 2)δ, (2.3) 8c4 = c1 4 + x [︂
Lemma 2
Lemma 2 [23] Let p ∈𝒫has power series expansion (2.1). Then, for μ ∈ℂ, |cm+k – μcmck| ≤2 max (1,|2μ – 1|), (2.5) |cm| ≤2 for m ≥1. (2.6)
Lemma 2 [23] Let p ∈𝒫has power series expansion (2.1). Then, for μ ∈ℂ, |cm+k – μcmck| ≤2 max (1,|2μ – 1|), (2.5) |cm| ≤2 for m ≥1. (2.6)
Lemma 3
Lemma 3 [24] Let p ∈𝒫has power series expansion (2.1), then |c3 – 2Bc1c2 + Dc1 3| ≤2, (2.7) if B ∈[0,1] and B(2B – 1) ≤D ≤B.
Lemma 3 [24] Let p ∈𝒫has power series expansion (2.1), then |c3 – 2Bc1c2 + Dc1 3| ≤2, (2.7) if B ∈[0,1] and B(2B – 1) ≤D ≤B.
Lemma 4
Lemma 4 [25] Let α, β, γ, and λ satisfying the conditions 0 < α < 1, 0 < λ < 1, and 8λ(1 – λ) [︃ (αβ – 2γ )2 + (α(λ + α) – β)2 ]︃ + α(1 –…
Lemma 4 [25] Let α, β, γ , and λ satisfying the conditions 0 < α < 1, 0 < λ < 1, and 8λ(1 – λ) [︃ (αβ – 2γ )2 + (α(λ + α) – β)2 ]︃ + α(1 – α)(β – 2λα)2 ≤4α2(1 – α)2λ(1 – λ). Let p ∈𝒫be given in (2.1), then the following inequality holds true ⃓⃓⃓⃓γ c4 1 + λc2 2 + 2αc1c3 – 3 2βc2 1c2 – c4 ⃓⃓⃓⃓≤2. (2.8) 3 Coefficients inequalities for the class 𝓚n–1,L
Theorem 1
Theorem 1 Let f ∈𝒦n–1,ℒbe of the form (1.1), then |a2| ≤ n 2(n + 1), |a3| ≤ n 6(n + 1), |a4| ≤ n 12(n + 1), |a5| ≤ n 20(n + 1). This result…
Theorem 1 Let f ∈𝒦n–1,ℒbe of the form (1.1), then |a2| ≤ n 2(n + 1), |a3| ≤ n 6(n + 1), |a4| ≤ n 12(n + 1), |a5| ≤ n 20(n + 1). This result is sharp for the functions given by f1(z) =
Theorem 2
Theorem 2 Let f ∈𝒦n–1,ℒbe of the form (1.1). Then, for γ ∈ℂ, |a3 – γ a2 2| ≤ n 6(n + 1) max (︃ 1, ⃓⃓⃓⃓ (3γ – 2)n 2(n + 1) ⃓⃓⃓⃓ )︃. This…
Theorem 2 Let f ∈𝒦n–1,ℒbe of the form (1.1). Then, for γ ∈ℂ, |a3 – γ a2 2| ≤ n 6(n + 1) max (︃ 1, ⃓⃓⃓⃓ (3γ – 2)n 2(n + 1) ⃓⃓⃓⃓ )︃ . This result is shown to be sharp in (3.2).
Corollary 1
Corollary 1 Let f ∈𝒦n–1,ℒ. Then, |a3 – a2 2| ≤ n 6(n + 1). (3.18) This result is shown to be sharp in (3.2).
Corollary 1 Let f ∈𝒦n–1,ℒ. Then, |a3 – a2 2| ≤ n 6(n + 1). (3.18) This result is shown to be sharp in (3.2).
Theorem 3
Theorem 3 Let f ∈𝒦n–1,ℒwith the series expansion (1.1). Then, |a4 – a2a3| ≤ n 12(n + 1). (3.19) This result is shown to be sharp in (3.3).
Theorem 3 Let f ∈𝒦n–1,ℒwith the series expansion (1.1). Then, |a4 – a2a3| ≤ n 12(n + 1). (3.19) This result is shown to be sharp in (3.3).
Theorem 4
Theorem 4 If f ∈𝒦n–1,ℒis given by (1.1), then |ℋ2,2(f )| = |a2a4 – a2 3| ≤ n2 36(n + 1)2. (3.21) This result is shown to be sharp in (3.2).
Theorem 4 If f ∈𝒦n–1,ℒis given by (1.1), then |ℋ2,2(f )| = |a2a4 – a2 3| ≤ n2 36(n + 1)2 . (3.21) This result is shown to be sharp in (3.2).
Corollary 2
Corollary 2 If f ∈𝒦5,ℒis given by (1.1), then |ℋ2,2(f )| ≤1 49 ≈0.0204. The result is sharp for the function f4(z) given by (3.2) with n =…
Corollary 2 If f ∈𝒦5,ℒis given by (1.1), then |ℋ2,2(f )| ≤1 49 ≈0.0204. The result is sharp for the function f4(z) given by (3.2) with n = 5, i.e., f4(z) = z + 1 7z3 + 1 1092z13 + ··· . 3.4 Third Hankel
Theorem 5
Theorem 5 If f ∈𝒦n–1,ℒis given by (1.1), then |ℋ3,1(f )| ≤43n3 + 33n2 2160(n + 1)3. (3.22)
Theorem 5 If f ∈𝒦n–1,ℒis given by (1.1), then |ℋ3,1(f )| ≤43n3 + 33n2 2160(n + 1)3 . (3.22)
Corollary 3
Corollary 3 If f ∈𝒦5,ℒis given by (1.1), then |ℋ3,1(f )| ≤ 291 20580 ≈0.0141.
Corollary 3 If f ∈𝒦5,ℒis given by (1.1), then |ℋ3,1(f )| ≤ 291 20580 ≈0.0141.
Function classes studied:
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