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
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