Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
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
Related Papers