THREE VALUE RANGES FOR SYMMETRIC SELF-MAPPINGS OF THE UNIT DISC.

Let D be the unit disc and z₀ ϵ D. We determine the value range {f(z₀) | f ϵ R≥}, where R≥ is the set of holomorphic functions f : D → D with f(0) = 0 and f'(0) ≥ 0 that have only real coefficients in their power series expansion around 0, and the smaller set {f(z₀) | f ϵ R≥, f is typically real}. Furthermore, we describe a third value range {f(z₀) | f ϵ I}, where I consists of all univalent self-mappings of the upper half-plane H with hydrodynamical normalization which are symmetric with respect to the imaginary axis. [ABSTRACT FROM AUTHOR]