Criteria for Bounded Valence of Harmonic Mappings

Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{|z|\rightarrow 1} |S(f)(z)| (1-|z|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Huusko, Juha-Matti [verfasserIn] Martín, María J.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2017

Schlagwörter:	Bounded valence criterion Harmonic mapping Pre-Schwarzian derivative Schwarzian derivative

Anmerkung:	© Springer-Verlag Berlin Heidelberg 2017

Übergeordnetes Werk:	Enthalten in: Computational methods and function theory - Berlin : Springer, 2001, 17(2017), 4 vom: 28. März, Seite 603-612
Übergeordnetes Werk:	volume:17 ; year:2017 ; number:4 ; day:28 ; month:03 ; pages:603-612

Links:	Volltext

DOI / URN:	10.1007/s40315-017-0197-z

Katalog-ID:	SPR036889474

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520			\|a Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic.
650		4	\|a Bounded valence criterion \|7 (dpeaa)DE-He213
650		4	\|a Harmonic mapping \|7 (dpeaa)DE-He213
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650		4	\|a Schwarzian derivative \|7 (dpeaa)DE-He213
700	1		\|a Martín, María J. \|4 aut
773	0	8	\|i Enthalten in \|t Computational methods and function theory \|d Berlin : Springer, 2001 \|g 17(2017), 4 vom: 28. März, Seite 603-612 \|w (DE-627)392236362 \|w (DE-600)2157081-4 \|x 2195-3724 \|7 nnns
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856	4	0	\|u https://dx.doi.org/10.1007/s40315-017-0197-z \|z lizenzpflichtig \|3 Volltext
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912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
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912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
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912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
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912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
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912			\|a GBV_ILN_2015
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912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
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912			\|a GBV_ILN_2057
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912			\|a GBV_ILN_2061
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912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2070
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912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
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912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2119
912			\|a GBV_ILN_2122
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912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
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912			\|a GBV_ILN_2188
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912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
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matchkey_str	article:21953724:2017----::rtraobuddaecohr
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publishDate	2017
allfields	10.1007/s40315-017-0197-z doi (DE-627)SPR036889474 (SPR)s40315-017-0197-z-e DE-627 ger DE-627 rakwb eng Huusko, Juha-Matti verfasserin aut Criteria for Bounded Valence of Harmonic Mappings 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2017 Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. Bounded valence criterion (dpeaa)DE-He213 Harmonic mapping (dpeaa)DE-He213 Pre-Schwarzian derivative (dpeaa)DE-He213 Schwarzian derivative (dpeaa)DE-He213 Martín, María J. aut Enthalten in Computational methods and function theory Berlin : Springer, 2001 17(2017), 4 vom: 28. März, Seite 603-612 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:17 year:2017 number:4 day:28 month:03 pages:603-612 https://dx.doi.org/10.1007/s40315-017-0197-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 17 2017 4 28 03 603-612
spelling	10.1007/s40315-017-0197-z doi (DE-627)SPR036889474 (SPR)s40315-017-0197-z-e DE-627 ger DE-627 rakwb eng Huusko, Juha-Matti verfasserin aut Criteria for Bounded Valence of Harmonic Mappings 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2017 Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. Bounded valence criterion (dpeaa)DE-He213 Harmonic mapping (dpeaa)DE-He213 Pre-Schwarzian derivative (dpeaa)DE-He213 Schwarzian derivative (dpeaa)DE-He213 Martín, María J. aut Enthalten in Computational methods and function theory Berlin : Springer, 2001 17(2017), 4 vom: 28. März, Seite 603-612 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:17 year:2017 number:4 day:28 month:03 pages:603-612 https://dx.doi.org/10.1007/s40315-017-0197-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 17 2017 4 28 03 603-612
allfields_unstemmed	10.1007/s40315-017-0197-z doi (DE-627)SPR036889474 (SPR)s40315-017-0197-z-e DE-627 ger DE-627 rakwb eng Huusko, Juha-Matti verfasserin aut Criteria for Bounded Valence of Harmonic Mappings 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2017 Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. Bounded valence criterion (dpeaa)DE-He213 Harmonic mapping (dpeaa)DE-He213 Pre-Schwarzian derivative (dpeaa)DE-He213 Schwarzian derivative (dpeaa)DE-He213 Martín, María J. aut Enthalten in Computational methods and function theory Berlin : Springer, 2001 17(2017), 4 vom: 28. März, Seite 603-612 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:17 year:2017 number:4 day:28 month:03 pages:603-612 https://dx.doi.org/10.1007/s40315-017-0197-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 17 2017 4 28 03 603-612
allfieldsGer	10.1007/s40315-017-0197-z doi (DE-627)SPR036889474 (SPR)s40315-017-0197-z-e DE-627 ger DE-627 rakwb eng Huusko, Juha-Matti verfasserin aut Criteria for Bounded Valence of Harmonic Mappings 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2017 Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. Bounded valence criterion (dpeaa)DE-He213 Harmonic mapping (dpeaa)DE-He213 Pre-Schwarzian derivative (dpeaa)DE-He213 Schwarzian derivative (dpeaa)DE-He213 Martín, María J. aut Enthalten in Computational methods and function theory Berlin : Springer, 2001 17(2017), 4 vom: 28. März, Seite 603-612 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:17 year:2017 number:4 day:28 month:03 pages:603-612 https://dx.doi.org/10.1007/s40315-017-0197-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 17 2017 4 28 03 603-612
allfieldsSound	10.1007/s40315-017-0197-z doi (DE-627)SPR036889474 (SPR)s40315-017-0197-z-e DE-627 ger DE-627 rakwb eng Huusko, Juha-Matti verfasserin aut Criteria for Bounded Valence of Harmonic Mappings 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2017 Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. Bounded valence criterion (dpeaa)DE-He213 Harmonic mapping (dpeaa)DE-He213 Pre-Schwarzian derivative (dpeaa)DE-He213 Schwarzian derivative (dpeaa)DE-He213 Martín, María J. aut Enthalten in Computational methods and function theory Berlin : Springer, 2001 17(2017), 4 vom: 28. März, Seite 603-612 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:17 year:2017 number:4 day:28 month:03 pages:603-612 https://dx.doi.org/10.1007/s40315-017-0197-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 17 2017 4 28 03 603-612
language	English
source	Enthalten in Computational methods and function theory 17(2017), 4 vom: 28. März, Seite 603-612 volume:17 year:2017 number:4 day:28 month:03 pages:603-612
sourceStr	Enthalten in Computational methods and function theory 17(2017), 4 vom: 28. März, Seite 603-612 volume:17 year:2017 number:4 day:28 month:03 pages:603-612
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Bounded valence criterion Harmonic mapping Pre-Schwarzian derivative Schwarzian derivative
isfreeaccess_bool	false
container_title	Computational methods and function theory
authorswithroles_txt_mv	Huusko, Juha-Matti @@aut@@ Martín, María J. @@aut@@
publishDateDaySort_date	2017-03-28T00:00:00Z
hierarchy_top_id	392236362
id	SPR036889474
language_de	englisch
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author	Huusko, Juha-Matti
spellingShingle	Huusko, Juha-Matti misc Bounded valence criterion misc Harmonic mapping misc Pre-Schwarzian derivative misc Schwarzian derivative Criteria for Bounded Valence of Harmonic Mappings
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topic_title	Criteria for Bounded Valence of Harmonic Mappings Bounded valence criterion (dpeaa)DE-He213 Harmonic mapping (dpeaa)DE-He213 Pre-Schwarzian derivative (dpeaa)DE-He213 Schwarzian derivative (dpeaa)DE-He213
topic	misc Bounded valence criterion misc Harmonic mapping misc Pre-Schwarzian derivative misc Schwarzian derivative
topic_unstemmed	misc Bounded valence criterion misc Harmonic mapping misc Pre-Schwarzian derivative misc Schwarzian derivative
topic_browse	misc Bounded valence criterion misc Harmonic mapping misc Pre-Schwarzian derivative misc Schwarzian derivative
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title	Criteria for Bounded Valence of Harmonic Mappings
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title_full	Criteria for Bounded Valence of Harmonic Mappings
author_sort	Huusko, Juha-Matti
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author_browse	Huusko, Juha-Matti Martín, María J.
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title_sort	criteria for bounded valence of harmonic mappings
title_auth	Criteria for Bounded Valence of Harmonic Mappings
abstract	Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. © Springer-Verlag Berlin Heidelberg 2017
abstractGer	Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. © Springer-Verlag Berlin Heidelberg 2017
abstract_unstemmed	Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. © Springer-Verlag Berlin Heidelberg 2017
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title_short	Criteria for Bounded Valence of Harmonic Mappings
url	https://dx.doi.org/10.1007/s40315-017-0197-z
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up_date	2024-07-03T20:12:18.287Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR036889474</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230328165939.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201007s2017 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s40315-017-0197-z</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR036889474</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s40315-017-0197-z-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Huusko, Juha-Matti</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Criteria for Bounded Valence of Harmonic Mappings</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2017</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer-Verlag Berlin Heidelberg 2017</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that %$\limsup _{\|z\|\rightarrow 1} \|S(f)(z)\| (1-\|z\|^2)^2 < 2%$, then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that %$\limsup _{\|z\|\rightarrow 1} \|f''(z)/f'(z)\|\, (1-\|z\|^2)< 1%$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Bounded valence criterion</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Harmonic mapping</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Pre-Schwarzian derivative</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Schwarzian derivative</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Martín, María J.</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Computational methods and function theory</subfield><subfield code="d">Berlin : Springer, 2001</subfield><subfield code="g">17(2017), 4 vom: 28. März, Seite 603-612</subfield><subfield code="w">(DE-627)392236362</subfield><subfield code="w">(DE-600)2157081-4</subfield><subfield code="x">2195-3724</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:17</subfield><subfield code="g">year:2017</subfield><subfield code="g">number:4</subfield><subfield code="g">day:28</subfield><subfield code="g">month:03</subfield><subfield code="g">pages:603-612</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://dx.doi.org/10.1007/s40315-017-0197-z</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_SPRINGER</subfield></datafield><datafield tag="912" 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Criteria for Bounded Valence of Harmonic Mappings

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