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Fekete-Szegö problem for certain subclass of quasi-convex functions. (English) Zbl 1119.30009

Summary: For \(0\leq\alpha<1\), let \({\mathcal Q}_\alpha\) be the class of functions \(f\) which are normalised analytic and univalent in \({\mathcal D}=\{z:|z|<1 \}\) satisfying the condition \[ \operatorname{Re}\left\{\frac{\alpha(z^2f''(z))'} {g'(z)}+\frac{(zf'(z))'} {g'(z)}\right\}>0, \] where \(g\) is a normalised convex function. For \(f\in{\mathcal Q}_\alpha\), sharp bounds are obtained for the Feketo-Szegö functional \(|a_3-\mu a_2^2|\) when \(\mu\) is real.

MSC:

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