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.

用\（{{\ cal P} _ {\ log}} \）表示所有非常数Pick函数f的集合，其对数导数f'/ f也属于Pick类。令\（{\ cal U}（{\ rm {\ Lambda}}）\）为函数族z·f（z），其中\（f \ in {{\ cal P} _ {\ log}} \），并且f在Λ≔[1，+√）上是全纯的。\（{\ cal U}（{\ rm {\ Lambda}}）\）中函数的重要示例是经典的多对数\（L {i_ \ alpha}（z）：= \ sum \ nolimits_ {k = 1} ^ \ infty {{z ^ k}} / {k ^ \ alpha} \），α≥0；见[5]（2015）。

在本说明中，我们证明每个\（\ varphi \ in {\ cal U}（{\ rm {\ Lambda}} \）都是星形的，即φ映射Λ中包含原点的原点的每个圆域一个到一个星形区域。此外，我们表明，对于某些常数p = p（f，在{{\ cal P} _ {\ log}} \）中的每个非恒定函数\（f \ in在上半平面上属于Hardy空间H _p）> 1，除非˚F正比于一些功能（一- ž）^{- θ}与一个εℝ和0 < θ1.≤最后，我们在存在用于其一个实数值函数υ导出充分必要条件（f中\ {{\ CAL P} _ {\日志}} \）\使得υ（X）= LIM _ε几乎所有x ∊ _↓0 lim f（x + iε）。