Universal convexity and universal starlikeness of polylogarithms

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