Fekete-Szegő problem for close-to-convex functions with respect to a certain convex function dependent on a real parameter. (English) Zbl 1348.30007
Summary: Given \(\alpha \in [0, 1]\), let \(h_{\alpha}(z):=z/(1- \alpha z)\), \(z \in \mathbb D:= \{z \in \mathbb C: |z| < 1\}\). An analytic standardly normalized function \(f\) in \(\mathbb D\) is called close-to-convex with respect to \(h_{\alpha}\) if there exists \(\delta \in (-\pi/2, \pi/2)\) such that \(\mathrm{Re}\{e^{i \delta}zf'(z)/h_{\alpha}(z)\} > 0\), \(z \in \mathbb D\). For the class \(\mathcal C(h_{\alpha})\) of all close-to-convex functions with respect to \(h_{\alpha}\), the Fekete-Szegö problem is studied.
MSC:
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
Keywords:
Fekete-Szegő problem; close-to-convex functions; close-to-convex functions with argument \(\delta\); close-to-convex functions with respect to a convex function; functions of bounded turningReferences:
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