The region of values of the system {?(z1), …, ?(zn)} in the class of typically real functions. II

The paper studies the region of values D_m,1(T) of the system {ƒ(z₁), ƒ(z₂), …, ƒ(zm), ƒ(r)}, m e 1, where z_j (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a₂z²+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set D_m,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication, the region of values of ƒ(z_m) in the subclass of functions from the class T with prescribed values ƒ(z_k) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005.