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Universal convexity and universal starlikeness of polylogarithms

Authors: Andrew Bakan, Stephan Ruscheweyh and Luis Salinas
Journal: Proc. Amer. Math. Soc. 143 (2015), 717-729
MSC (2010): Primary 30C45, 30H10; Secondary 44A15
Published electronically: October 29, 2014
MathSciNet review: 3283658
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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 $\alpha \geq 1$.

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Additional Information

Andrew Bakan
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 01601, Ukraine

Stephan Ruscheweyh
Affiliation: Institut für Mathematik, Universität Würzburg, 97074 Würzburg, Germany

Luis Salinas
Affiliation: Departamento de Informática, UTFSM, Valparaíso, Chile

Keywords: Convex functions, polylogarithms, universally convex functions, universally starlike functions, Pick functions
Received by editor(s): December 19, 2012
Received by editor(s) in revised form: May 10, 2013
Published electronically: October 29, 2014
Additional Notes: The second and third authors acknowledge support from FONDECYT, Grant 1100805, from Basal Project FB0821 CCTVal-Centro Científico Tecnológico de Valparaíso, and from Anillo Project ACT119. This work was completed while the first author was visiting Würzburg University, supported by the German Academic Exchange Service (DAAD, grant 322-A/11/05274)
Communicated by: Walter Van Assche
Article copyright: © Copyright 2014 American Mathematical Society