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.

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.