Universal convexity and universal starlikeness of polylogarithms

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 $\alpha \geq 1$.