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The Fekete-Szegő theorem for close-to-convex functions of the class \(K_{sh}(\alpha,\beta)\). (English) Zbl 1017.30007
In this paper the author introduces and studies the following class of normalized close-to-convex functions in the unit disc \(D\). For \(0\leq \alpha<1\) and \(0<\beta\leq 1\), let \(K_{sh}(\alpha,\beta)\) be the class of normalized close-to-convex functions defined in the open unit disc \(D\) by \[ \left|\arg\left(\frac{zf'(z)}{g(z)}\right)\right|\leq \frac{\pi\alpha}{2}, \] such that \(g\in S^*(\beta)\) where \(S^*(\beta)\) is the class of analytic normalized starlike functions of order \(\beta\), that is \[ Re\left(\frac{zg'(z)}{g(z)}\right)>\beta. \] In the paper the author obtains sharp bounds for the Fekete-Szegö functional \(|a_3-\mu a_2^2|\) when \(\mu\) is real and \(f(z)=z+a_2z^2+a_3z^3+\dots\) is in the class \(K_{sh}(\alpha,\beta)\).

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