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The set of values of initial coefficients of bounded univalent functions. (English. Russian original) Zbl 1115.30302
Russ. Math. 42, No. 8, 12-19 (1998); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1998, No. 8, 14-21 (1998).
Introduction: Let $$S$$ be a class of regular functions $$f(z)= z+a_2z^2+\cdots$$ which are univalent in the unit circle $$E=\{z: |z|< 1\}$$; $$S^M$$ be a class of functions $$f\in S$$ which satisfy in $$E$$ the condition $$|f(z)|< M$$, $$M>1$$.
In the theory of univalent (schlicht) functions, one of the central places is occupied by the coefficient problem, which consists of a description of the set of values of the system $$V_n= \{a_2,\dots, a_n\}$$. In particular, the monographs by K. I. Babenko [The theory of extremal problems for univalent functions of class $$S$$. Proc. Steklov Inst. Math. 101 (1972), translation from Tr. Mat. Inst. Steklova 101 (1972; Zbl 0283.30020)] and A. S. Schaeffer and D. C. Spencer [Coefficient regions for schlicht functions. American Mathematical Society Colloquium Publ. 35, New York: AMS (1950; Zbl 0066.05701)] are devoted to the investigation of the sets $$V_n$$. In these monographs the analytical and topological properties of the boundary $$\partial V_n$$ of the set $$V_n$$ are studied. It was shown in [K. I. Babenko (loc. cit.)] that if $$A_1$$ is the set of singularities of $$\partial V_n$$ and $$x\in A_1$$, then the function $$f(z)$$, which supplies the point $$x$$, is an algebraic function. In [A. S. Schaeffer and D. C. Spencer (loc. cit.)] a complete description of the set $$V_3$$ in class $$S$$ was obtained.
Recently, O. Tammi and D. V. Prokhorov [see: Reachable set methods in extremal problems for univalent functions, Saratov: Saratov University Publishing House (1993; Zbl 0814.30016)] actively studied the sets $$V_n$$. D. V. Prokhorov obtained a complete description of the boundary $$\partial V_4(M)$$ in the class $$S_R^M$$ of typically real functions $$f\in S$$, which satisfy in the circle $$E$$ the inequality $$|f(z)|< M$$.
In the present article, by means of application of the methods of optimization for controllable systems generated by the Loewner equation, a complete description of the boundary $$\partial U_4(M)$$ of the set $$U_4(M)$$, which is a projection of the set $$V_4(M)$$ to the space $$(a_2, a_3, \operatorname{Re} a_4)$$ in class $$S^M$$, is obtained. Prokhorov’s hypothesis of the character of angular points and a curve which connects these points and is an edge of the boundary hypersurface was confirmed. The results of the article strengthen the hypothesis on two functionals in the class $$S$$. In particular, a local extremum of two functionals can be supplied by functions which send the unit circle to the plane with two cuts along the real axis.
##### MSC:
 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods