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The Fekete-Szegö problem for a subclass of close-to-convex functions. (English) Zbl 1021.30014
Summary: Let \({\mathcal C}_1(\beta)\) be the class of normalized functions \(f\), which are analytic in the open unit disk \({\mathcal U}\), given by the power series: \(f(z)= z+ \sum_{n=2}^\infty a_nz^n\), and satisfy the inequality: \[ \text{Re}\Biggl\{ \frac{zf'(z)} {\varphi(z)} e^{i\beta} \Biggr\}> 0 \qquad \biggl( z\in{\mathcal U};\;-\frac\pi 2< \beta< \frac\pi 2\biggr), \] for some normalized univalent and convex function \(\varphi\). In this paper we solve the Fekete-Szegő problem for the family: \[ {\mathcal C}_1:= \cup_\beta{\mathcal C}_1(\beta) \qquad \biggl( -\frac\pi 2< \beta< \frac\pi 2 \biggr) \] by proving that \[ \max_{f\in{\mathcal C}_1}|a_3- \lambda a_2^2|= \begin{cases} \frac 53- \frac{9\lambda}4 &(0\leq \lambda\leq \frac 29),\\ \frac 23+ \frac 1{9\lambda} &(\frac 29 \leq \lambda\leq \frac 23),\\ \frac 56 &(\frac 23\leq \lambda\leq 1). \end{cases} \] .

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