Universally starlike and Pick functions

TL;DR

This paper proves that functions derived from a special class of Pick functions are universally starlike, explores their Hardy space membership, and characterizes boundary limits of these functions.

Contribution

It establishes universal starlikeness of a broad class of functions related to Pick functions and characterizes their boundary behavior and Hardy space properties.

Findings

Functions in the class are universally starlike.

Non-constant functions in the class belong to Hardy spaces for some p>1.

Characterization of boundary limits of functions in the class.

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 $Li_\alpha(z)$ $:=$ $\sum_{k=1}^{\infty} z^k / k^\alpha$ for $\alpha \geq 0$. In this paper we prove that every $\varphi \in \mathcal{U}(\Lambda)$ is universally starlike, i.e., $\varphi$ 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 $0 < \theta \leq 1$. 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 $x \in \mathbb{R}$.