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Three value ranges for symmetric self-mappings of the unit disc


Authors: Julia Koch and Sebastian Schleißinger
Journal: Proc. Amer. Math. Soc. 145 (2017), 1747-1761
MSC (2010): Primary 30C55, 30C80
DOI: https://doi.org/10.1090/proc/13350
Published electronically: October 26, 2016
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Abstract: Let $ \mathbb{D}$ be the unit disc and $ z_0\in \mathbb{D}.$ We determine the value range $ \{f(z_0)\,\vert\, f\in \mathcal {R}^\geq \}$, where $ \mathcal {R}^\geq $ is the set of holomorphic functions $ f:\mathbb{D}\to \mathbb{D}$ with $ f(0)=0$ and $ f'(0)\geq 0$ that have only real coefficients in their power series expansion around 0, and the smaller set $ \{f(z_0)\,\vert\, f\in \mathcal {R}^\geq ,$$ \text {$f$\ is typically real}\}.$

Furthermore, we describe a third value range $ \{ f(z_0) \,\vert\, f \in \mathcal {I}\}$, where $ \mathcal {I}$ consists of all univalent self-mappings of the upper half-plane $ \mathbb{H}$ with hydrodynamical normalization which are symmetric with respect to the imaginary axis.


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  • [Dur83] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • [GB92] V. V. Goryaĭnov and I. Ba, Semigroup of conformal mappings of the upper half-plane into itself with hydrodynamic normalization at infinity, Ukraïn. Mat. Zh. 44 (1992), no. 10, 1320-1329 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 44 (1992), no. 10, 1209-1217 (1993). MR 1201130, https://doi.org/10.1007/BF01057676
  • [Gor86] V. V. Goryaĭnov, Semigroups of conformal mappings, Mat. Sb. (N.S.) 129(171) (1986), no. 4, 451-472, 591 (Russian). MR 842395
  • [Gor15] V. V. Goryaĭnov, Evolution families of conformal mappings with fixed points and the Löwner-Kufarev equation, Mat. Sb. 206 (2015), no. 1, 39-68 (Russian, with Russian summary); English transl., Sb. Math. 206 (2015), no. 1-2, 33-60. MR 3354961, https://doi.org/10.4213/sm8276
  • [KS16] Julia Koch and Sebastian Schleißinger, Value ranges of univalent self-mappings of the unit disc, J. Math. Anal. Appl. 433 (2016), no. 2, 1772-1789. MR 3398791, https://doi.org/10.1016/j.jmaa.2015.08.068
  • [Pfr16] D. Pfrang, Der Wertebereich typisch-reeller schlichter beschränkter Funktionen, Master's thesis, University of Würzburg, 2016.
  • [Pro92] D. V. Prokhorov, The set of values of bounded univalent functions with real coefficients, Theory of functions and approximations, Part 1 (Russian) (Saratov, 1990), Izdat. Saratov. Univ., Saratov, 1992, pp. 56-60 (Russian). MR 1335945
  • [PS16] D. V. Prokhorov and K. Samsonova, A Description Method in the Value Region Problem, Complex Analysis and Operator Theory (2016).
  • [Rob35] M. S. Robertson, On the coefficients of a typically-real function, Bull. Amer. Math. Soc. 41 (1935), no. 8, 565-572. MR 1563142, https://doi.org/10.1090/S0002-9904-1935-06147-6
  • [Rog32] Werner Rogosinski, Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), no. 1, 93-121 (German). MR 1545292, https://doi.org/10.1007/BF01186552
  • [Rog34] W. Rogosinski, Zum Schwarzschen Lemma, Jahresber. Dtsch. Math.-Ver. 44 (1934), 258-261.
  • [RS14] Oliver Roth and Sebastian Schleißinger, Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation, Bull. Lond. Math. Soc. 46 (2014), no. 5, 1099-1109. MR 3262210, https://doi.org/10.1112/blms/bdu054
  • [SS82] Maria Szapiel and Wojciech Szapiel, Extreme points of convex sets. IV. Bounded typically real functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), no. 1-2, 49-57 (English, with Russian summary). MR 667386
  • [Tam78] Olli Tammi, Extremum problems for bounded univalent functions, Lecture Notes in Mathematics, Vol. 646, Springer-Verlag, Berlin-New York, 1978. MR 0492191
  • [Win88] Gerhard Winkler, Extreme points of moment sets, Math. Oper. Res. 13 (1988), no. 4, 581-587. MR 971911, https://doi.org/10.1287/moor.13.4.581

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Additional Information

Julia Koch
Affiliation: Universität Würzburg, Institut für Mathematik, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
Email: julia.d.koch@mathematik.uni-wuerzburg.de

Sebastian Schleißinger
Affiliation: Università di Roma “Tor Vergata”, Dipartimento di Matematica, 00133 Roma, Italy
Email: schleiss@mat.uniroma2.it

DOI: https://doi.org/10.1090/proc/13350
Keywords: Value ranges, radial Loewner equation, chordal Loewner equation, univalent functions, typically real functions
Received by editor(s): February 16, 2016
Received by editor(s) in revised form: June 20, 2016, and June 25, 2016
Published electronically: October 26, 2016
Additional Notes: The second author was supported by the ERC grant “HEVO - Holomorphic Evolution Equations” No. 277691.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society

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