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Sharp bounds of the third Hankel determinant for classes of univalent functions with bounded turning. (English) Zbl 1513.30071

Let \(\mathcal{A}\) denote the class of functions \(f(z)=z+a_2z^2+a_3z^3+\dots\) that are analytic in the unit disk \(D=\{z\in\mathbb{C}:|z|<1\}\). The authors consider the third order case of the Hankel determinant \[H_3(1)=a_3(a_2a_4-a_2^2)-a_4(a_4-a_2a_3)+a_5(a_3-a_2^2)\] for two classes of univalent functions with bounded turning.
They study the class \(\mathcal{R}\subset\mathcal{A}\) of univalent functions satisfying \[\mathrm{Re}f'(z)>0,\ \ z\in D,\] and the class \(\mathcal{R}_1\subset\mathcal{A}\) satisfying \[\mathrm{Re}(f'(z)+zf''(z))>0,\ \ z\in D.\] The main results are the following theorems: if \(f\in\mathcal{R}\) then \[|H_3(1)|\le\frac{207}{540}=0,38333\dots\]
and if \(f\in\mathcal{R}_1\) then \[|H_3(1)|\le\frac{3537}{129600}=0,02729\dots .\]

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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