Universally starlike and Pick functions
Denote by $\mathcal{P}_{\log}$ the set of all non-constant Pick functions $f$ whose logarithmic derivatives $f^{\, \prime}/f$ also belong to the Pick class. Let $\mathcal{U}(Λ)$ be the family of functions $z\cdot f(z)$, where $f \in\mathcal{P}_{\log}$ and $f$ is holomorphic on $Λ:=\mathbb{C}\setminus [1, +\infty)$. Important examples of functions in $\mathcal{U}(Λ)$ are the classical polylogarithms $Li_α(z)$ $:=$ $\sum_{k=1}^{\infty} z^k / k^α$ for $α\geq 0$. In this paper we prove that every $φ\in \mathcal{U}(Λ)$ 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 $f \in \mathcal{P}_{\log}$ 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 $a \in \mathbb{R}$ and $0 < θ\leq 1$. Finally we derive a necessary and sufficient condition on a real-valued function $v$ for which there exists $f \in \mathcal{P}_{\log}$ such that $v (x) = \lim_{\varepsilon \to 0} \mathrm{Im} f (x + i \varepsilon)$ for almost all $x \in \mathbb{R}$.