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Monotonicity theorems for analytic functions centered at infinity


Author: Galatia Cleanthous
Journal: Proc. Amer. Math. Soc. 142 (2014), 3545-3551
MSC (2010): Primary 30C25, 30C35, 30C75
DOI: https://doi.org/10.1090/S0002-9939-2014-12084-3
Published electronically: June 23, 2014
MathSciNet review: 3238429
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the family of analytic functions centered at infinity with Laurent expansion $ f(z)=cz+c_{0}+\sum _{j=1}^{\infty }c_{j}z^{-j}.$ We prove some monotonicity theorems involving geometric quantities such as diameter, radius and length.


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  • [1] Rauno Aulaskari and Huaihui Chen, Area inequality and $ Q_p$ norm, J. Funct. Anal. 221 (2005), no. 1, 1-24. MR 2124895 (2005k:30066), https://doi.org/10.1016/j.jfa.2004.12.007
  • [2] Dimitrios Betsakos and Stamatis Pouliasis, Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius, Canad. Math. Bull. 56 (2013), no. 2, 241-250. MR 3043051, https://doi.org/10.4153/CMB-2011-189-8
  • [3] Robert B. Burckel, Donald E. Marshall, David Minda, Pietro Poggi-Corradini, and Thomas J. Ransford, Area, capacity and diameter versions of Schwarz's lemma, Conform. Geom. Dyn. 12 (2008), 133-152. MR 2434356 (2010j:30050), https://doi.org/10.1090/S1088-4173-08-00181-1
  • [4] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494 (85j:30034)
  • [5] James A. Jenkins, A uniqueness result in conformal mapping, Proc. Amer. Math. Soc. 22 (1969), 324-325. MR 0241619 (39 #2958)
  • [6] James A. Jenkins, A uniqueness result in conformal mapping. II, Proc. Amer. Math. Soc. 85 (1982), no. 2, 231-232. MR 652448 (83h:30020), https://doi.org/10.2307/2044287
  • [7] Christian Pommerenke, Univalent functions, with a chapter on quadratic differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. MR 0507768 (58 #22526)
  • [8] Albert Pfluger, On a uniqueness theorem in conformal mapping, Michigan Math. J. 23 (1976), no. 4, 363-365 (1977). MR 0442207 (56 #593)
  • [9] Albert Pfluger, On the diameter of planar curves and Fourier coefficients, Z. Angew. Math. Phys. 30 (1979), no. 2, 305-314 (English, with German summary). MR 535988 (81b:42024), https://doi.org/10.1007/BF01601942
  • [10] G. Pólya and G. Szegö, Problems and Theorems in Analysis. Vol. I, Springer-Verlag, Berlin, 1972 (first edition in German, 1925). MR 0344042
  • [11] G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Theory of functions, zeros, polynomials, determinants, number theory, geometry. Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Die Grundlehren der Mathematischen Wissenschaften, Band 216, Springer-Verlag, New York, 1976. MR 0396134 (53 #2)
  • [12] Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001)

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Additional Information

Galatia Cleanthous
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
Email: gkleanth@math.auth.gr

DOI: https://doi.org/10.1090/S0002-9939-2014-12084-3
Keywords: Meromorphic functions, conformal mapping, logarithmic capacity, monotonicity theorems, univalent functions, Schwarz Lemma.
Received by editor(s): October 19, 2012
Received by editor(s) in revised form: October 24, 2012, and October 29, 2012
Published electronically: June 23, 2014
Additional Notes: The author would like to thank D. Betsakos, her thesis advisor, for his help, and the Cyprus State Scholarship Foundation for its support.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society

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