Normality Properties of the Family f(nz): n ∈ N

Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introdu...
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Autor*in:	Nevo, Shahar [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2001

Schlagwörter:	Normal family

Anmerkung:	© Heldermann Verlag 2001

Übergeordnetes Werk:	Enthalten in: Computational methods and function theory - Berlin : Springer, 2001, 1(2001), 2 vom: Nov., Seite 375-386
Übergeordnetes Werk:	volume:1 ; year:2001 ; number:2 ; month:11 ; pages:375-386

Links:	Volltext

DOI / URN:	10.1007/BF03320997

Katalog-ID:	SPR036823015

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520			\|a Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m.
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allfields	10.1007/BF03320997 doi (DE-627)SPR036823015 (SPR)BF03320997-e DE-627 ger DE-627 rakwb eng Nevo, Shahar verfasserin aut Normality Properties of the Family f(nz): n ∈ N 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Heldermann Verlag 2001 Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. Normal family (dpeaa)DE-He213 Enthalten in Computational methods and function theory Berlin : Springer, 2001 1(2001), 2 vom: Nov., Seite 375-386 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:1 year:2001 number:2 month:11 pages:375-386 https://dx.doi.org/10.1007/BF03320997 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_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_2118 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 1 2001 2 11 375-386
spelling	10.1007/BF03320997 doi (DE-627)SPR036823015 (SPR)BF03320997-e DE-627 ger DE-627 rakwb eng Nevo, Shahar verfasserin aut Normality Properties of the Family f(nz): n ∈ N 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Heldermann Verlag 2001 Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. Normal family (dpeaa)DE-He213 Enthalten in Computational methods and function theory Berlin : Springer, 2001 1(2001), 2 vom: Nov., Seite 375-386 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:1 year:2001 number:2 month:11 pages:375-386 https://dx.doi.org/10.1007/BF03320997 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_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_2118 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 1 2001 2 11 375-386
allfields_unstemmed	10.1007/BF03320997 doi (DE-627)SPR036823015 (SPR)BF03320997-e DE-627 ger DE-627 rakwb eng Nevo, Shahar verfasserin aut Normality Properties of the Family f(nz): n ∈ N 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Heldermann Verlag 2001 Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. Normal family (dpeaa)DE-He213 Enthalten in Computational methods and function theory Berlin : Springer, 2001 1(2001), 2 vom: Nov., Seite 375-386 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:1 year:2001 number:2 month:11 pages:375-386 https://dx.doi.org/10.1007/BF03320997 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_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_2118 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 1 2001 2 11 375-386
allfieldsGer	10.1007/BF03320997 doi (DE-627)SPR036823015 (SPR)BF03320997-e DE-627 ger DE-627 rakwb eng Nevo, Shahar verfasserin aut Normality Properties of the Family f(nz): n ∈ N 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Heldermann Verlag 2001 Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. Normal family (dpeaa)DE-He213 Enthalten in Computational methods and function theory Berlin : Springer, 2001 1(2001), 2 vom: Nov., Seite 375-386 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:1 year:2001 number:2 month:11 pages:375-386 https://dx.doi.org/10.1007/BF03320997 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_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_2118 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 1 2001 2 11 375-386
allfieldsSound	10.1007/BF03320997 doi (DE-627)SPR036823015 (SPR)BF03320997-e DE-627 ger DE-627 rakwb eng Nevo, Shahar verfasserin aut Normality Properties of the Family f(nz): n ∈ N 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Heldermann Verlag 2001 Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. Normal family (dpeaa)DE-He213 Enthalten in Computational methods and function theory Berlin : Springer, 2001 1(2001), 2 vom: Nov., Seite 375-386 (DE-627)392236362 (DE-600)2157081-4 2195-3724 nnns volume:1 year:2001 number:2 month:11 pages:375-386 https://dx.doi.org/10.1007/BF03320997 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_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_2118 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 1 2001 2 11 375-386
language	English
source	Enthalten in Computational methods and function theory 1(2001), 2 vom: Nov., Seite 375-386 volume:1 year:2001 number:2 month:11 pages:375-386
sourceStr	Enthalten in Computational methods and function theory 1(2001), 2 vom: Nov., Seite 375-386 volume:1 year:2001 number:2 month:11 pages:375-386
format_phy_str_mv	Article
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topic_facet	Normal family
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container_title	Computational methods and function theory
authorswithroles_txt_mv	Nevo, Shahar @@aut@@
publishDateDaySort_date	2001-11-01T00:00:00Z
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language_de	englisch
fullrecord	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR036823015</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230328165923.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201007s2001 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/BF03320997</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR036823015</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)BF03320997-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">Nevo, Shahar</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Normality Properties of the Family f(nz): n ∈ N</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2001</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">© Heldermann Verlag 2001</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Normal family</subfield><subfield code="7">(dpeaa)DE-He213</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">1(2001), 2 vom: Nov., Seite 375-386</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:1</subfield><subfield code="g">year:2001</subfield><subfield code="g">number:2</subfield><subfield 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author	Nevo, Shahar
spellingShingle	Nevo, Shahar misc Normal family Normality Properties of the Family f(nz): n ∈ N
authorStr	Nevo, Shahar
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topic_title	Normality Properties of the Family f(nz): n ∈ N Normal family (dpeaa)DE-He213
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topic_unstemmed	misc Normal family
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title	Normality Properties of the Family f(nz): n ∈ N
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title_full	Normality Properties of the Family f(nz): n ∈ N
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title_sort	normality properties of the family f(nz): n ∈ n
title_auth	Normality Properties of the Family f(nz): n ∈ N
abstract	Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. © Heldermann Verlag 2001
abstractGer	Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. © Heldermann Verlag 2001
abstract_unstemmed	Abstract A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on DE, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Qm-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain DE, where E = E(S) ⊂ D satisfies ED(m) = ∅. m is said to be the degree of Ƒ. (Here ED(m) is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write fn(z) = f(nz). Then the family Π(f) = fn: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Qm-normal for any value of m. © Heldermann Verlag 2001
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title_short	Normality Properties of the Family f(nz): n ∈ N
url	https://dx.doi.org/10.1007/BF03320997
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doi_str	10.1007/BF03320997
up_date	2024-07-03T19:46:19.080Z
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ind1=" " ind2=" "><subfield code="a">GBV_ILN_4326</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4333</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4334</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4335</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4336</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4338</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4393</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4700</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">1</subfield><subfield code="j">2001</subfield><subfield code="e">2</subfield><subfield code="c">11</subfield><subfield code="h">375-386</subfield></datafield></record></collection>
score	7.400154

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Normality Properties of the Family f(nz): n ∈ N

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Zugang & Verfügbarkeit

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