Abstract

In this note we prove that every φ ∈ U(Λ) is universally starlike, i.e., φ maps every circular domain in Λ containing the origin one-to-one onto a starlike domain. Furthermore, we show that every non-constant function f ∈ P log belongs to the Hardy space H p on the upper half-plane for some constant p = p(f ) > 1, unless f is proportional to some function (a−z) −θ with a ∈ R and 0 < θ ≤ 1. Finally we derive a necessary and sufficient condition on a real-valued function v for which there exists f ∈ P log such that v(x) = lim ε↓0 lim f (x + iε) for almost all x ∈ R.