On the homogeneous combination \(a_ 4+\mu a^ 3_ 2\) for bounded real univalent functions
Let \(S_ R(b)\), \(0<b\leq 1\), denote the class of functions of the form \(f(z)=b(z+\sum^{\infty}_{n=2}a_ nz^ n),\) \(a_ n\in {\mathbb{R}}\), \(n=2,3,...\), holomorphic, univalent and satisfying the condition \(| f(z)| <1\) for \(| z| <1.\) In the paper the functional \(a_ 4+\mu a^ 3_ 2,\) where \(\mu\) is an arbitrary real parameter, is considered in the class \(S_ R(b)\). In consequence of the investigations carried out, the functional is maximized in six disjoint sets in the plane (b,\(\mu)\). Two Grunsky-type estimations: the power inequality [\textit{O. Tammi}, Extremum problems for bounded univalent functions (1978; Zbl 0375.30006)] and the Jokinen inequalities [\textit{O. Jokinen}, Ann. Acad. Sci. Fenn., Ser. A. I, Diss. 41, 52 p. (1982; Zbl 0492.30010)] are used in the proof.
scientific article; zbMATH DE number 3899231
scientific article