Zaprawa, Paweł On the Fekete-Szegő problem for classes of bi-univalent functions. (English) Zbl 1300.30035 Bull. Belg. Math. Soc. - Simon Stevin 21, No. 1, 169-178 (2014). Let \(\mathcal{S}\) denote the class of univalent functions in the unit disk \(\mathbb{D}\) of the form \(f(z) = z + a_2 z^2 + \cdots\). Following M. Lewin [Proc. Am. Math. Soc. 18, 63–68 (1967; Zbl 0158.07802)], the author denotes by \(\sigma\) the class of bi-univalent functions, i.e., the class of functions \(f\) for which both \(f\) and \(f^{-1}\) are univalent in \(\mathbb{D}\). Next, let \(\mathcal{ST}(\alpha,\varphi)\) and \(\mathcal{M}(\alpha,\varphi)\) denote the classes of functions \(f\) for which \[ \frac{zf'(z)}{f(z)}+\alpha\frac{z^2f''(z)}{f(z)}\prec \varphi(z)\quad \text{and}\quad \frac{wg'(w)}{g(w)}+\alpha\frac{w^2g''(w)}{g(w)}\prec \varphi(w), \] where \(g=f^{-1}\), and \[ \begin{split} (1-\alpha)\frac{zf'(z)}{f(z)}+\alpha\left(1+\frac{zf''(z)}{f(z)}\right)\prec \varphi(z)\quad \text{and}\\ (1-\alpha) \frac{wg'(w)}{g(w)}+\alpha\left(1+\frac{wg''(w)}{g(w)}\right)\prec \varphi(w),\end{split} \] respectively. In the defined classes \(\mathcal{ST}(\alpha,\varphi)\) and \(\mathcal{M}(\alpha,\varphi)\), the Fekete-Szegő problem is solved. Reviewer: Stanisława Kanas (Rzeszów) Cited in 62 Documents MSC: 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions of one complex variable Keywords:bi-univalent functions; bi-starlike functions; bi-convex functions; Fekete-Szegő functional Citations:Zbl 0158.07802 × Cite Format Result Cite Review PDF Full Text: Euclid