Universally starlike and Pick functions

Abstract

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 $\mathcal{U}(\Lambda )$ be the family of functions $z\cdot f(z)$, where $f\in {\mathcal{P}}_{\log }$ and $f$ is holomorphic on $\Lambda :=\mathbb{C}\setminus [1,+\infty )$. Important examples of functions in $\mathcal{U}(\Lambda )$ are the classical polylogarithms $L{i}_{\alpha }(z)$ $:=$ ${\sum }_{k=1}^{\infty }{z}^k/k^{\alpha }$ for $\alpha \ge 0$. In this paper we prove that every $\phi \in \mathcal{U}(\Lambda )$ is universally starlike, i.e., $\phi$ maps every circular domain in $\Lambda$ 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 $(a-z{)}^{-\theta }$ with $a\in \mathbb{R}$ and $0<\theta \le 1$. 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 $v(x)={\lim }_{\epsilon \to 0}\mathrm{Im}f(x+i\epsilon )$ for almost all $x\in \mathbb{R}$.

40 pages