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Abstract
In this paper we improve the bounds of the third order Hankel determinant for two classes of univalent functions with bounded turning. The bounds are not sharp, but the sharp ones are 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. Then, for any real numbers and such that, where and We will also use the following, almost forgotten…
Lemma 1. Let $\omega(z) = c_1 z + c_2 z^2 + \cdots$ be a Schwarz function. Then, for any real numbers $\mu$ and $\nu$ such that $(\mu, \nu) \in D_1 \cup D_2$ , where
$$D_1 = \left\{ (\mu, \nu) : |\mu| \le \frac{1}{2}, -1 \le \nu \le 1 \right\}$$
and
$$D_2 = \left\{ (\mu, \nu) : \frac{1}{2} \le |\mu| \le 2, \, \frac{4}{27} (|\mu| + 1)^3 - (|\mu| + 1) \le \nu \le 1 \right\},$$
$the\ following\ sharp\ estimate\ holds$
$$\left| c_3 + \mu c_1 c_2 + \nu c_1^3 \right| \le 1.$$
We will also use the following, almost forgotten result of Carleson ([3]).
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 $f \in \mathcal{R}$ is of the form $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ . Then
$$|H_3(1)| \le \frac{207}{540} = 0.38333\dots$$
Theorem 2 · coeff
Theorem 2. Let and is of the form. Then
Theorem 2. Let $f \in \mathcal{R}_1$ and is of the form $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ . Then
$$|H_3(1)| \le \frac{3537}{129600} = 0.02729\dots$$
Function classes studied:
Coefficient bounds & claims (4)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_3(1) ≤ 207/540 for class R [Theorem 1]
coefficient_bound
H_3(1) ≤ 3537/129600 for class R1 [Theorem 2]
function_family
Class R: f in A with Re f'(z) > 0 for z in D (bounded turning)
function_family
Class R1: f in A with Re{f'(z) + z f''(z)} > 0 for z in D
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