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Coefficient estimates for analytic functions concerned with Hankel determinant. (English) Zbl 1221.30031

Summary: For the coefficients of analytic functions \(f(z)\) defined by the conditions \(f(0)=f'(0)-1=0\) and
\[ \text{Re\,}\bigg(\alpha\frac{f(z)}{z}+\beta f'(z)\bigg)>\gamma \]
for some complex parameters \(\alpha\) and \(\beta\) satisfying \(\alpha+n\beta\not=0\), \(n=1,2,\dots\), for some real number \(\gamma\), \(0\leq \gamma<1\), where \(\text{Re\,}(\alpha+\beta)>\gamma\) in the open unit disk \(\mathbb U\), the upper bounds of the generalized functional \( \big| a^{}_n a^{}_{n+2}-\mu a^2_{n+1}\big|\) concerning the second Hankel determinant \(H_2(n)\) for all \(n=1,2,\ldots\) and for some real number \(\mu\), are discussed. Furthermore, by means of these results, the same things for starlike functions of order \(\gamma\) in \(\mathbb U\), \(1/2\leq \gamma<1\), are also considered.

MSC:

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