Abstract
Let f be analytic in D = {z ∈C : |z| < 1}, and be given by
f(z) = z + ∞
n=2 anzn. We give sharp bounds for the second Hankel
determinant, some Toeplitz, and some Hermitian-Toeplitz determinants
of functions in the class of Ozaki close-to-convex functions, together
with a sharp bound for the Zalcman functional J2,3(f).
Mathematics Subject Classification. Primary 30C45, 30C55.
Results & Lemmas (14)
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.
Theorem 1.1.
Theorem 1.1. Let f ∈F(λ), λ > −1/2, and be given by (1). Then, for n ≥2 |an| ≤1 n! n k=2 (k + 2λ −1). (4) The inequality is sharp when…
Theorem 1.1. Let f ∈F(λ), λ > −1/2, and be given by (1). Then, for n ≥2 |an| ≤1 n! n k=2 (k + 2λ −1). (4) The inequality is sharp when f(z) = fλ(z) := 1 2λ 1 (1 −z)2λ −1
Lemma 2.1.
Lemma 2.1. ([10]) Let p ∈P, and be given by (6), then |pn| ≤2, when n ≥2. Also p2 −μ 2 p2 1 ≤max 2, 2|μ −1| = 2, 0 ≤μ ≤2, 2|μ −1|,…
Lemma 2.1. ([10]) Let p ∈P, and be given by (6), then |pn| ≤2, when n ≥2. Also p2 −μ 2 p2 1 ≤max{2, 2|μ −1|} = 2, 0 ≤μ ≤2, 2|μ −1|, elsewhere. (7)
Lemma 2.2.
Lemma 2.2. ([11]) If p ∈P, and is given by (6), then for μ ∈C, and 1 ≤k ≤ n −1 |pn −μpkpn−k| ≤2 max 1, |2μ −1|.
Lemma 2.2. ([11]) If p ∈P, and is given by (6), then for μ ∈C, and 1 ≤k ≤ n −1 |pn −μpkpn−k| ≤2 max{1, |2μ −1|}.
Lemma 2.3.
Lemma 2.3. ([17]) Suppose that p ∈P, with coefficients given by (6), and p1 ≥0. Then, for some complex-valued y with |y| ≤1, and some…
Lemma 2.3. ([17]) Suppose that p ∈P, with coefficients given by (6), and p1 ≥0. Then, for some complex-valued y with |y| ≤1, and some complex- valued ζ with |ζ| ≤1 2p2 = p2 1 + y(4 −p2 1), 4p3 = p3 1 + 2(4 −p2 1)p1y −p1(4 −p2 1)y2 + 2(4 −p2 1)(1 −|y|2)ζ.
Lemma 2.4.
Lemma 2.4. Suppose f ∈F0(λ), and is given by (1). Then a2 = 1 4p1(1 + 2λ), a3 = 1 12(1 + 2λ)
Lemma 2.4. Suppose f ∈F0(λ), and is given by (1). Then a2 = 1 4p1(1 + 2λ), a3 = 1 12(1 + 2λ)
Theorem 3.1.
Theorem 3.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then |H2(2)(f)| ≤(1 + 2λ)2(17 −10λ) 192(3 −2λ). The inequality is sharp.
Theorem 3.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then |H2(2)(f)| ≤(1 + 2λ)2(17 −10λ) 192(3 −2λ) . The inequality is sharp.
Corollary 3.2.
Corollary 3.2. If f ∈C, then |H2(2)(f)| ≤1 8, proved in [13], and if f ∈FO(1), i.e., satisfying (2), then |H2(2)(f)| ≤21 64, proved in [5].…
Corollary 3.2. If f ∈C, then |H2(2)(f)| ≤1 8, proved in [13], and if f ∈FO(1), i.e., satisfying (2), then |H2(2)(f)| ≤21 64, proved in [5]. 4. Toeplitz determinants In this section, we extend the results in [2] for f ∈C to f ∈FO(λ). We first define the function f1 ∈FO(λ) for z ∈D, which serves as the extreme function for all the following results: f1(z) := 1 2iλ
Theorem 4.1.
Theorem 4.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then for n ≥2 |T2(n)(f)| ≤
Theorem 4.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then for n ≥2 |T2(n)(f)| ≤
Theorem 4.2.
Theorem 4.2. If f ∈FO(λ), 1/2 ≤λ ≤1, then |T3(1)(f)| ≤1 18(4 + 3λ + 2λ2)(7 + 6λ + 8λ2). The inequality is sharp when f = f1.
Theorem 4.2. If f ∈FO(λ), 1/2 ≤λ ≤1, then |T3(1)(f)| ≤1 18(4 + 3λ + 2λ2)(7 + 6λ + 8λ2). The inequality is sharp when f = f1.
Theorem 4.3.
Theorem 4.3. If f ∈FO(λ), 1/2 ≤λ ≤1, then |T3(2)(f)| ≤ 1 864(1 + 2λ)3(9 + 5λ + 2λ2)(25 + 17λ + 10λ2). The inequality is sharp when f = f1.
Theorem 4.3. If f ∈FO(λ), 1/2 ≤λ ≤1, then |T3(2)(f)| ≤ 1 864(1 + 2λ)3(9 + 5λ + 2λ2)(25 + 17λ + 10λ2). The inequality is sharp when f = f1.
Theorem 5.1.
Theorem 5.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then T3,1(f) ≤ ⎧ ⎨ ⎩ 1, λ ∈ 1/2, ( √ 153 −5)/8 , 1 + 1 18(1 + 2λ)2(4λ2 + 5λ −8),
Theorem 5.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then T3,1(f) ≤ ⎧ ⎨ ⎩ 1, λ ∈ 1/2, ( √ 153 −5)/8 , 1 + 1 18(1 + 2λ)2(4λ2 + 5λ −8),
Corollary 5.2.
Corollary 5.2. If f ∈C, then 0 ≤T3,1(f) ≤1, proved in [9], and if f ∈FO(1), then −49 320 ≤T3,1(f) ≤3 2. 6. The functional J2,3(f) We give…
Corollary 5.2. If f ∈C, then 0 ≤T3,1(f) ≤1, proved in [9], and if f ∈FO(1), then −49 320 ≤T3,1(f) ≤3 2. 6. The functional J2,3(f) We give the sharp upper bound for |J2,3(f)| when f ∈FO(λ).
Theorem 6.1.
Theorem 6.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then |J2,3(f)| ≤(1 + 2λ)(7 + 2λ)3/2 36 3(9 −4λ2). The inequality is sharp.
Theorem 6.1. If f ∈FO(λ), 1/2 ≤λ ≤1, then |J2,3(f)| ≤(1 + 2λ)(7 + 2λ)3/2 36 3(9 −4λ2) . The inequality is sharp.
Corollary 6.2.
Corollary 6.2. If f ∈C, then |J2,3(f)| ≤ 4 9 √ 3, proved in [4], and if f ∈FO(1), then |J2,3(f)| ≤ 9 4 √ 15, proved in [8].…
Corollary 6.2. If f ∈C, then |J2,3(f)| ≤ 4 9 √ 3, proved in [4], and if f ∈FO(1), then |J2,3(f)| ≤ 9 4 √ 15, proved in [8]. Acknowledgements The authors would like to thank the referee for his constructive comments
Function classes studied:
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