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Abstract
In this paper we improve the upper bound of the third order Hankel determinant for the class of Ozaki close-to-convex functions. The sharp bound is conjectured.
Results & Lemmas (4)
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. Let be a Schwarz function, i.e., be analytic in the unit dick and when and and be real numbers. If and, then. We will also need…
Lemma 1. Let $\omega(z) = c_1 z + c_2 z^2 + \cdots$ be a Schwarz function, i.e., be analytic in the unit dick and $|\omega(z)| < 1$ when $z \in \mathbb{D}$ and $\mu$ and $\nu$ be real numbers. If $\frac{1}{2} \leq |\mu| \leq 2$ and $\frac{4}{27}(|\mu|+1)^3 - (|\mu|+1) \leq \nu \leq 1$ , then $|c_3 + \mu c_1 c_2 + \nu c_1^3| \leq 1$ .
We will also need the following, almost forgotten result of Carleson ([2]).
Lemma 2
Lemma 2. Let be a Schwarz function. Then and.
Lemma 2. Let $\omega(z) = c_1 z + c_2 z^2 + \cdots$ be a Schwarz function. Then
$$|c_2| \le 1 - |c_1|^2$$
and $|c_4| \le 1 - |c_1|^2 - |c_2|^2$ .
Theorem 1 · coeff
Theorem 1. Let is of the form. Then
Theorem 1. Let $f \in \mathcal{F}$ is of the form $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ . Then
$$|H_3(1)| \le \frac{1}{8} = 0.125.$$
Theorem 2 · coeff
Theorem 2. Let and is of the form. Then
Theorem 2. Let $f \in \mathcal{G}$ and is of the form $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ . Then
$$|H_3(1)| \le \frac{17}{1080} = 0.01574\dots$$
Function classes studied:
Coefficient bounds & claims (4)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H3(1) ≤ 1/8 for class F [Theorem 1]
coefficient_bound
H3(1) ≤ 17/1080 for class G [Theorem 2]
function_family
Class F: f in A: Re(1 + z*f''(z)/f'(z)) > -1/2 for z in D (Ozaki close-to-convex functions)
function_family
Class G: f in A: Re(1 + z*f''(z)/f'(z)) < 3/2 for z in D
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