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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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