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Abstract
In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class $\mathcal{S}$ of univalent functions in the unit disc.
Results & Lemmas (1)
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Theorem 1
Theorem 1. For the class S we have, where and, where
Theorem 1. For the class S we have
$$|H_2(2)| \le A$$
, where $1 \le A \le \frac{11}{3} = 3,66...$
and
$$|H_3(1)| \le B$$
, where $\frac{4}{9} \le B \le \frac{32 + \sqrt{285}}{15} = 3.258796 \cdots$
Coefficient bounds & claims (8)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_2(2) = |a2*a4 - a3^2| ≤ 11/3 for class S [Theorem 1]
coefficient_bound
H_2(2) = |a2*a4 - a3^2| ≤ 1 for class S (sharp) [Theorem 1]
coefficient_bound
H_3(1) ≤ (32 + sqrt(285))/15 for class S [Theorem 1]
coefficient_bound
H_3(1) ≤ 4/9 for class S (sharp) [Theorem 1]
function_family
Class S: All analytic univalent functions f in unit disk D with f(z) = z + a2*z^2 + a3*z^3 + ...
function_family
Class S*: Starlike functions in D
function_family
Class U: f in A satisfying |(z/f(z))^2 * f'(z) - 1| < 1
function_family
Class K: Convex functions in D
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