## Universal convexity and universal starlikeness of polylogarithms

HTML articles powered by AMS MathViewer

- 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

- Gearoid de Barra,
*Measure theory and integration*, Ellis Horwood Series in Mathematics and its Applications, Ellis Horwood Ltd., Chichester; John Wiley & Sons, Inc., New York, 1981. MR**637464** - William F. Donoghue Jr.,
*Monotone matrix functions and analytic continuation*, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR**0486556** - Peter L. Duren,
*Univalent functions*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR**708494** - Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,
*Higher transcendental functions. Vol. I*, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR**698779** - Alexander B. Goncharov,
*Polylogarithms in arithmetic and geometry*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 374–387. MR**1403938** - Pedro Jodrá,
*On a connection between the polylogarithm function and the Bass diffusion model*, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.**464**(2008), no. 2099, 3081–3088. MR**2439315**, DOI 10.1098/rspa.2008.0196 - Frederick W. King,
*Hilbert transforms. Vol. 1*, Encyclopedia of Mathematics and its Applications, vol. 124, Cambridge University Press, Cambridge, 2009. MR**2542214** - Paul Koosis,
*Introduction to $H_p$ spaces*, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR**1669574** - A. Mukherjea and K. Pothoven,
*Real and functional analysis*, Mathematical Concepts and Methods in Science and Engineering, Vol. 6, Plenum Press, New York-London, 1978. MR**0492145** - John L. Lewis,
*Convexity of a certain series*, J. London Math. Soc. (2)**27**(1983), no. 3, 435–446. MR**697137**, DOI 10.1112/jlms/s2-27.3.435 - Walter Rudin,
*Real and complex analysis*, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR**0344043** - St. Ruscheweyh and T. Sheil-Small,
*Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture*, Comment. Math. Helv.**48**(1973), 119–135. MR**328051**, DOI 10.1007/BF02566116 - Stephan Ruscheweyh and Luis Salinas,
*Universally prestarlike functions as convolution multipliers*, Math. Z.**263**(2009), no. 3, 607–617. MR**2545859**, DOI 10.1007/s00209-008-0433-3 - Stephan Ruscheweyh, Luis Salinas, and Toshiyuki Sugawa,
*Completely monotone sequences and universally prestarlike functions*, Israel J. Math.**171**(2009), 285–304. MR**2520111**, DOI 10.1007/s11856-009-0050-9

## 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