Abstract
Let A denote the class of normalized analytic functions f in the open unit disk
defined as D := {z ∈C : |z| < 1} with f(0) = 0 and f′(0) = 1. A function f ∈A is
said to be starlike if f(D) is starlike domain. By using the Bernstein polynomial method to
obtain the required maximum estimate, we establish sharp upper bound for the third Hankel
determinant corresponding to the inverse coefficients of starlike univalent (i.e., one-to-one)
functions in the unit disk D.
Results & Lemmas (2)
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 1.1
Lemma 1.1. [13] Let and be given by (1.2) with. Then for some such that, and. 2. Main Result
Lemma 1.1. [13] Let $p \in \mathcal{P}$ and be given by (1.2) with $c_1 \geq 0$ . Then
$$2c_{2} = c_{1}^{2} + \gamma (4 - c_{1}^{2}),$$
$$4c_{3} = c_{1}^{3} + 2(4 - c_{1}^{2})c_{1}\gamma - (4 - c_{1}^{2})c_{1}\gamma^{2} + 2(4 - c_{1}^{2})(1 - |\gamma|^{2})\eta,$$
$$8c_{4} = c_{1}^{4} + (4 - c_{1}^{2})\gamma (c_{1}^{2}(\gamma^{2} - 3\gamma + 3) + 4\gamma)$$
$$-4(4 - c_{1}^{2})(1 - |\gamma|^{2})(c_{1}(\gamma - 1)\eta + \overline{\gamma}\eta^{2} - (1 - |\eta|^{2})\rho)$$
for some $\gamma, \eta, \rho$ such that $|\gamma| \leq 1$ , $|\eta| \leq 1$ and $|\rho| \leq 1$ .
2. Main Result
Theorem 2.1
Theorem 2.1. Let be given by (1.1). Then The inequality is sharp for the Koebe function.
Theorem 2.1. Let $f \in \mathcal{S}^*$ be given by (1.1). Then
$$|H(3,1)(f^{-1})| \le 1.$$
The inequality is sharp for the Koebe function.
Coefficient bounds & claims (2)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H(3,1)(f^{-1}) ≤ 1 for class S* (sharp) [Theorem 2.1]
function_family
Class S*: f in A such that Re(zf'(z)/f(z)) > 0 for z in D; class of starlike univalent functions