Let \(S^ M_ R\), \(M>1\), denote the class of functions \(f(z)=z+a_ 2z^ 2+\cdots\) holomorphic, univalent and bounded in the disc \(E=\{z\in\mathbb{C}:| z|<1\}\). O. Tammi [Extremum problems for bounded univalent functions. I, II, Lect. Notes Math. 646 (1978; Zbl 0375.30006) and 913 (1982; Zbl 0481.30020)] obtained partial results concerning the determination of the boundary \(\partial V_ 4(M)\) of the value set \(V_ 4(M)\) of the system \((a_ 2,a_ 3,a_ 4)\) of the coefficients \(a_ 2,a_ 3,a_ 4\) of a function \(f\in S^ M_ R\). Of course, each function \(f\) of this class is a typically-real function.
The author solves this problem entirely (Theorem 1). Moreover, it turns out (Theorem 2) that the boundary \(\partial V_ 4(M)\) of the set \(V_ 4(M)\) is a two-dimensional surface, smooth everywhere with the exception of two points corresponding to the Pick function and some curve joining these points. The partial results and hypotheses of O. Tammi have found their confirmation here.