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Abstract
The objective of this paper is to find the best possible upper bound of the third Hankel determinant for the inverse of convex functions.
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. If, is of the form (1.5) with, such that then where, for some, and such that, and.
Lemma 1.1. If $p \in \mathcal{P}$ , is of the form (1.5) with $c_1 \geq 0$ , such that $c_1 \in [0,2]$ then
$$2c_2 = c_1^2 + \nu \mu,$$
$$4c_3 = c_1^3 + 2c_1\nu\mu - c_1\nu\mu^2 + 2\nu\left(1 - |\mu|^2\right)\rho,$$
$$8c_4 = c_1^4 + 3c_1^2\nu\mu + \left(4 - 3c_1^2\right)\nu\mu^2 + c_1^2\nu\mu^3 + 4\nu\left(1 - |\mu|^2\right)\left(1 - |\rho|^2\right)\psi + 4\nu\left(1 - |\mu|^2\right)\left(c_1\rho - c\mu\rho - \bar{\mu}\rho^2\right),$$
where $\nu := 4 - c_1^2$ , for some $\mu$ , $\rho$ and $\psi$ such that $|\mu| \le 1$ , $|\rho| \le 1$ and $|\psi| \le 1$ .
Theorem 2.1
Theorem 2.1. If, then and the inequality is sharp for.
Theorem 2.1. If $f \in \mathcal{S}^c$ , then
$$|H_{3,1}(f^{-1})| \le \frac{1}{36}$$
and the inequality is sharp for $p_0(z) = (1+z^3)/(1-z^3)$ .
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}) ≤ 1/36 for class Sc (sharp) [Theorem 2.1]
function_family
Class Sc: Re(1 + z*f''(z)/f'(z)) > 0; convex univalent functions in the unit disk
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