Coefficients of holomorphic functions (Q1889947)

Denote by \(S\) the class of holomorphic univalent functions \(f\) in the disc \(E=\{z\in \mathbb{C}:|z|< 1\}\) of the form \[ f(z)= z+\sum^\infty_{n=2} a_n z^n \] and by \(S(M)\), and \(M> 1\), the subclasses of \(S\) of functions \(f\) satisfying the condition \(|f(z)|< M\) for \(z\in E\). This paper contains the interesting survey of various problems ,conjectures, results and research methods concerning mainly the classes \(S\) and \(S(M)\). Contents: Introduction. 1. Variational and parametric methods in extremal probles for univalent functions. 2.1. General problems on systems of functionals and coefficient systems. 2.2. Optimization methods in the theory of univalent functions (contains many results of the author). 2.3. The Jakubowski conjecture. 2.4. The Charzyński-Tammi conjecture. 2.5. Curves on the boundary hypersurface of the coefficient set. 3. The two-functional conjecture. 4.1. The Krzyż conjecture. 4.2. The generalized Krzyż conjecture. 4.3. Estimates of initial coefficients for univalent functions of the class \(B_S\). 4.4. Optimal control theory methods in extremal problems for the class \(B\). The bibliography consists of 77 references. It seems that the described subject one can widen studying e.g. the following papers: 1) \textit{D. V. Prokhorov}, Bounded univalent functions, chapter 8 of the book: Kühnau R. (ed.) Handbook of complex analysis: geometric function theory, Vol. 1, 207--228 (Amsterdam: North Holland) (2002; Zbl 1083.30014). 2) \textit{A. W. Goodman}, Univalent functions (Mariner Publ. Comp., Tampa) Vol. I, Chapter 4.3; Vol. II, Chapter 12.2 (1983; Zbl 1041.30500). 3) \textit{Z. Charzyński}, Sur les fonctions univalentes bornées, Rosprawy Mat. 2 (1953; Zbl 0053.04702). 4) \textit{Z. Charzyński} and \textit{W. Janowski}, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 4, 41--56 (1950; Zbl 0040.32802). 5) \textit{Z. J. Jakubowski}, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 54, 45--52 (2000; Zbl 0984.30011). 6) \textit{Z. J. Jakubowski}, Zesz. Nauk. Politech. Rzesz. 26, 5--30 (2002). 7) \textit{Z. J. Jakubowski}, \textit{A. Zielińska} and \textit{K. Zyskowska}, Ann. Pol. Math. 40, 193--206 (1983; Zbl 0531.30018). 8) \textit{A. Zielińska}, Demonstr. Math. 15, 381--396 (1982; Zbl 0514.30017). 9) \textit{K. Zyskowska}, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 35, 149--157 (1981; Zbl 0557.30015).