Abstract

A deep result of J. Lewis (1983) shows that the polylogarithms $ Li_\alpha (z)$ $ :=$ $ \sum _{k=1}^{\infty }z^k/k^\alpha $ map the open unit disk $ \mathbb{D} $ centered at the origin one-to-one onto convex domains for all $ \alpha \geq 0$. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain $ \Lambda := \mathbb{C}\setminus [1,\infty )$, introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that $ Li_\alpha (z)$ maps an arbitrary open disk or half-plane in $ \Lambda $ one-to-one onto a convex domain for every $ \alpha \geq 1$.