Universally starlike and Pick functions

Abstract

Denote by ${{\cal P}_{\log}}$ the set of all non-constant Pick functions f whose logarithmic derivatives f′/f also belong to the Pick class. Let ${\cal U}({\rm{\Lambda}})$ be the family of functions z · f(z), where $f \in {{\cal P}_{\log}}$ and f is holomorphic on Λ ≔ ℂ [1, + √). Important examples of functions in ${\cal U}({\rm{\Lambda}})$ are the classical polylogarithms $L{i_\alpha}(z): = \sum\nolimits_{k = 1}^\infty {{z^k}} /{k^\alpha}$ for α ≥ 0; see [5](2015).

In this note we prove that every $\varphi \in {\cal U}({\rm{\Lambda}})$ 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 $f \in {{\cal P}_{\log}}$ 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 υ for which there exists $f \in {{\cal P}_{\log}}$ such that υ (x) = lim_ε↓0 lim f(x + iε) for almost all x ∊ ℝ.