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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Universal convexity and universal starlikeness of polylogarithms
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by Andrew Bakan, Stephan Ruscheweyh and Luis Salinas PDF
Proc. Amer. Math. Soc. 143 (2015), 717-729 Request permission

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 $\alpha \geq 1$.
References
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Additional Information
  • Andrew Bakan
  • Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 01601, Ukraine
  • Email: andrew@bakan.kiev.ua
  • Stephan Ruscheweyh
  • Affiliation: Institut für Mathematik, Universität Würzburg, 97074 Würzburg, Germany
  • Email: ruscheweyh@mathematik.uni-wuerzburg.de
  • Luis Salinas
  • Affiliation: Departamento de Informática, UTFSM, Valparaíso, Chile
  • Email: luis.salinas@usm.cl
  • 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
  • © Copyright 2014 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 717-729
  • MSC (2010): Primary 30C45, 30H10; Secondary 44A15
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12262-3
  • MathSciNet review: 3283658