Fekete-Szegő problems for quasi-subordination classes
In 1970 \textit{M. Robertson} [Bull. Am. Math. Soc. 76, 1--9 (1970; Zbl 0191.09101)] introduced the notion of quasi-subordination. The function \(f\) is quasi-subordinated to \(g\), written as \(f \prec_q q\), if there exist the analytic functions \(\varphi\) and \(\omega\) satisfying \(|\varphi(z)| \leq 1\), \(\omega(0) = 0\), and \(|\omega(z)|<1\) for \(|z| <1\) and such that \(f(z) = \varphi(z)g(\omega(z))\). For the case \(\varphi(z)\equiv z\) the quasi-subordination reduces to the subordination, and for \(\omega(z) \equiv z\) to the majorization. Based on the above definition the authors define the classes \(\mathcal{M}_q(\alpha, \phi)\) and \(\mathcal{L}_q(\alpha, \varphi)\) as the classes of analytic functions \(f\) with \(f(0)=f'(0)-1=0\), and satisfying \[ (1-\alpha)\frac{zf'(z)}{f(z)}+\alpha\left(1+\frac{zf''(z)}{f'(z)}\right)-1\prec_q\varphi(z)-1, \] \[ \left(\frac{zf'(z)}{f(z)}\right)^\alpha\left(1+\frac{zf''(z)}{f'(z)}\right)^{1-\alpha}-1\prec_q\varphi(z)-1, \] respectively. For the above defined class the Fekete-Szegő functional is estimated.
scientific article; zbMATH DE number 6116340
scientific article