Abstract
In this paper we determine the upper bounds of the Hankel determinants of special type $H_{2}(3)(f)$ and $H_{2}(4)(f)$ for the class of univalent functions and for the class $\mathcal{U}$ defined by \[ \mathcal{U}=\left\{ f\in\mathcal{A} : \left|\left[\frac{z}{f(z)}\right]^2 f'(z)-1 \right|<1,\, z\in{\mathbb D} \right\}, \] where $\mathcal{A}$ is the class of functions analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+\cdots$.
Results & Lemmas (3)
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 · coeff
Theorem 1. Let. Then - (a) if, and the result is sharp due to the function. (b) for every.
Theorem 1. Let $f \in \mathcal{U}$ . Then
- (a) $|H_2(3)(f)| \le 1$ if $a_2 = 0$ , and the result is sharp due to the function $f(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \cdots$ . (b) $|H_2(3)(f)| \le 1.4846575 \dots$ for every $f \in \mathcal{U}$ .
Theorem 2 · coeff
Theorem 2. Let and. Then and the estimate is sharp due to the function.
Theorem 2. Let $f \in \mathcal{U}$ and $a_2 = 0$ . Then $|H_2(4)(f)| \le 1$ and the estimate is sharp due to the function $f(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + z^7 + \cdots$ .
Theorem 3 · coeff
Theorem 3. Let is given by (1). Then - (a) if; (b) for every.
Theorem 3. Let $f \in \mathcal{S}$ is given by (1). Then
- (a) $|H_2(3)(f)| \le 2.02757...$ if $a_2 = 0$ ; (b) $|H_2(3)(f)| \le 4.8986977...$ for every $f \in \mathcal{S}$ .
Coefficient bounds & claims (7)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_2(3)(f) = a_3*a_5 - a_4^2 ≤ 1 for class U (sharp) [Theorem 1(a)]
coefficient_bound
H_2(3)(f) = a_3*a_5 - a_4^2 ≤ 1.4846575 for class U [Theorem 1(b)]
coefficient_bound
H_2(4)(f) = a_4*a_6 - a_5^2 ≤ 1 for class U (sharp) [Theorem 2]
coefficient_bound
H_2(3)(f) = a_3*a_5 - a_4^2 ≤ 2.02757 for class S [Theorem 3(a)]
coefficient_bound
H_2(3)(f) = a_3*a_5 - a_4^2 ≤ 4.8986977 for class S [Theorem 3(b)]
function_family
Class U: f in A: |(z/f(z))^2 f'(z) - 1| < 1, z in D
function_family
Class S: class of all univalent functions in A normalized by f(0)=0, f'(0)=1
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