The region of values of a system of functionals in the class of typically real functions

Let TR be the class of functions f(z) = z + ∑n=2∞ cnzn that are regular and typically real in the disk E = {z : |z| < 1}. For this class, the region of values of the system {f(z0), f(r)} for z0 ∈ ℝ, r, ∈ (-1, 1) is studied. The sets Dr = (f(z0) : f ∈ TRf(r) = a} for -1 ≤ r ≤ 1 and Δr = {(c2, c3) : f ∈ TR, - f(-r) = a} for 0 < r ≤ 1 are found, where a ∈ (r(1 + r)-2, r(1 - r)-2) is an arbitrary fixed number. Bibliography: 11 titles. © 1998 Plenum Publishing Corporation.