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題名 半純函數的唯一性
Some Results on the Uniqueness of Meromorphic Functions
作者 陳耿彥
Chen, Keng-Yan
貢獻者 陳天進
Chen, Ten-Ging
陳耿彥
Chen, Keng-Yan
關鍵詞 值分佈理論
半純函數
value distribution theory
meromorphic function
日期 2007
上傳時間 17-Sep-2009 13:47:35 (UTC+8)
摘要 在這篇論文裡，我們利用值分佈的理論來探討半純函數的共值與唯一性的問題，本文包含了以下的結果：將Jank與Terglane有關三個A類中的半純函數唯一性的結果推廣到任意q個半純函數的情形；證明了C. C. Yang的一個猜測；建構了一類半純函數恰有兩個虧值，而且算出它們的虧格；將
Nevanlinna 五個值的定理推廣至兩個半純函數部分共值的情形；探討純函數
與其導數的共值問題；最後，證明了兩個半純函數共四個值且重數皆不同的定
理。
In this thesis, we study the sharing value problems and the
uniqueness problems of meromorphic functions in the theory of value distribution. In fact, this thesis contains the following results: We generalize a unicity condition of three meromorphic functions given by Jank and Terglane in class A to the case of arbitrary q meromorphic functoins. An elementary proof of a conjecture of C. C. Yang is provided. We construct a class of meromorphic functions with exact two deficient values and their deficiencies are explicitly computed. We generalize the Nevanlinna`s five-value theorem to the cases that two meromorphic functions partially share either five or more values, or five or
more small functions. In each case, we formulate a way to measure how far these two meromorphic functions are from sharing either values or small functions, and use this measurement to prove a uniqueness theorem. Also, we prove some uniqueness theorems on entire functions that share a pair of values (a,-a) with their derivatives, which are reformulations of some important results about uniqueness of entire functions that share values with their derivatives. Finally, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ share four distinct values a_1, a_2, a_3, a_4 DM such that each a_i-point is either a (p,q)-fold or (q,p)-fold point of f and g, then (p,q) is either (1,2) or (1,3) and f, g are in some particular forms.
參考文獻 [1] W. W. Adams and E. G. Straus, Non-Archimedian analytic
functions taking the same values at the same points,
Ill. J. Math., 15 (1971), 418-424.
[2] G. Brosch, Eindeutigkeitssatze fur meromorphe
funktionen, Thesis, Technical University of Aachen,
1989.
[3] J. Clunie, On integral and meromorphic functions,
J. London Math. Soc., 36 (1962), 17-27.
[4] C. T. Chuang and C. C. Yang, Fixed points and
factorization theory of meromorphic functions, Peking
Univ. Press, 1988.
[5] W. Doeringer, Exceptional value of differential
polynomial, Pacific J. Math., 98 (1982), 55-62.
[6] G. Frank and W. Ohlenroth, Meromorphe funktionen, die
mit einer ihrer ableitungen werte teilen, Complex
Variables, 6 (1986), 23-37.
[7] F. Gross, Factorizatioin of meromorphic functions, U.
S. Government Printing Office, Washington, D. C.,1972.
[8] G. G. Gundersen, Meromorphic functions that share three
or four values, J. London Math. Soc., 20 (1979),
457-466.
[9] G. G. Gundersen, Meromorphic functions that share finite
values with their derivative, J. Math. Anal. Appl.,
75 (1980), 441-446.
[10] G. G. Gundersen, Meromorphic functions that share four
values, Transactions of the American Mathematical
Society, 277(2) (1983), 545-567.
[11] G. G. Gundersen and L. Z. Yang, Entire functions that
share one value with one or two of their derivatives,
J. Math. Anal. Appl., 223 (1998), 88-95.
[12] W. K. Hayman, Meromorphic functions, Clarendon Press,
Oxford, 1964.
[13] D. Hans and S. Gerald, Zur charakterisierung von
polynomen durch ihre Null-und Einsstellen, Arch.
Math., 48 (1987), 337-342.
[14] G. Jank and N. Terglane, Meromorphic functions sharing
three values, Math. Pannonica, 2 (1990), 37-46.
[15] P. Li, Entire functions that share one value with their
linear differential polynomials, Kodai Math. J., 22
(1999), 446-457.
[16] P. Li and C. C. Yang, Uniqueness theorems on entire
functions and their derivatives, J. Math. Anal. Appl.,
253 (2001), 50-57.
[17] Y. Li and J. Qiao, The uniqueness of meromorphic
functions concerning small functions, Sci. China Ser.
A, 43(6) (2000), 581-590.
[18] E. Mues, Meromorphic functions sharing four values,
Complex Variables, 12 (1989), 169-179.
[19] E. Mues, G. Jank and L. Volkmann, Meromorphe
funktionen, die mit ihrer ersten und zweiten ableitung
einen endichen wert teilen, Complex Variables Theory
Appl. 6(1986), 51-71.
[20] E. Mues and N. Steinmetz, Meromorphe funktionen, die
mit ihrer abelitung werte teilen, Manuscripta Math.
29 (1979), 195-206.
[21] H. Milloux, Les fonctions meromorphes et leurs
derivees, Paris, 1940.
[22] S. S. Miller, Complex analysis: Proceedings of the SUNY
Brockport Conference, Dekker, New York and Basel,
1978, p.169.
[23] T. T. Moh, On a certain group structure for
polynomials, Proc. Amer. Math. Soc., 82 (1981),
183-187.
[24] K. Ninno and M. Ozawa, Deficiencies of an entire
algebroid function, Kodai Math. Sem. Rep., 22 (1970),
98-113.
[25] R. Nevanlinna, Le theoreme de Picard-Borel et la
theorie des fonctions meromorphes, Gauthiers-Villars,
Paris, 1929.
[26] R. Nevanlinna, Einige eindueutigkeitssatze in der
theorie der mermorphen funktionen, Acta Math., 48
(1926), 367-391.
[27] E. Picard. Memoire sur les fonctions entieres, Ann.
Ecole. Norm., 9(1880), 145-166.
[28] G. Polya. On an integral function of an integral
function, J. London Math. Soc., 1(1926), p.12.
[29] L. Ruble and C. C. Yang, Values shared by entire
functions and their derivatives, Complex Analysis,
Kentucky, 1976 (Berlin),Springer-Verlag, 1977, 101-103.
[30] M. Reinders, Eindeutigkeitssatze fur meromprphe
Funktionen, die vier Werte teilen, PhD thesis,
Universitat Hannover, 1990.
[31] M. Reinders, Eindeutigkeitssatze fur meromorphe
funktionen, die vier werte teilen, Mitt. Math. Sem.
Giessen, 200 (1991), 15-38.
[32] M. Reinders, A new example of meromorphic functions
sharing four values and a uniqueness theorem, Complex
Variables, 18 (1992), 213-221.
[33] N. Steinmetz, Eine Verallgemeinerung des zweiten
Nevanlinnaschen Hauptsatzes, J. Reine Angew. Math.,
368 (1986) 134-141.
[34] S. P. Wang, On meromorphic functions that share four
values, J. Math. Anal. Appl., 173 (1993), 359-369.
[35] H. X. Yi and C. C. Yang, Uniqueness theory of
meromorphic functions, Pure and Applied Math.
Monographs No. 32, Science Press, Beijing, 1995.
[36] C. C. Yang, Some problems on polynomyals and
transcendental entire functions, Adv. Math.
(a Chinese Journal), 13 (1984), 1-3.
[37] C. C. Yang. On deficiencies of differential
polynomials, Math. Z., 116 (1970), 197-204.
[38] L. Yang, Value distribution theory, Berlin Heidelberg:
Springer-Verlag, Beijing:Science Press, 1993.
[39] L. Z. Yang, Solution of a differential equation and its
applications, Kodai Math. J. 22 (1990), No.3, 458-464.
[40] Q. D. Zhang, A uniqueness theorem for meromorphic
functions with respect to slowly growing functions,
Acta Math. Sinica, 36(6) (1993), 826-833.
描述 博士
國立政治大學
應用數學研究所
93751501
96
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0093751501
資料類型 thesis
dc.contributor.advisor 陳天進zh_TW
dc.contributor.advisor Chen, Ten-Gingen_US
dc.contributor.author (Authors) 陳耿彥zh_TW
dc.contributor.author (Authors) Chen, Keng-Yanen_US
dc.creator (作者) 陳耿彥zh_TW
dc.creator (作者) Chen, Keng-Yanen_US
dc.date (日期) 2007en_US
dc.date.accessioned 17-Sep-2009 13:47:35 (UTC+8)-
dc.date.available 17-Sep-2009 13:47:35 (UTC+8)-
dc.date.issued (上傳時間) 17-Sep-2009 13:47:35 (UTC+8)-
dc.identifier (Other Identifiers) G0093751501en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/32582-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 93751501zh_TW
dc.description (描述) 96zh_TW
dc.description.abstract (摘要) 在這篇論文裡，我們利用值分佈的理論來探討半純函數的共值與唯一性的問題，本文包含了以下的結果：將Jank與Terglane有關三個A類中的半純函數唯一性的結果推廣到任意q個半純函數的情形；證明了C. C. Yang的一個猜測；建構了一類半純函數恰有兩個虧值，而且算出它們的虧格；將
Nevanlinna 五個值的定理推廣至兩個半純函數部分共值的情形；探討純函數
與其導數的共值問題；最後，證明了兩個半純函數共四個值且重數皆不同的定
理。		zh_TW
dc.description.abstract (摘要) In this thesis, we study the sharing value problems and the
uniqueness problems of meromorphic functions in the theory of value distribution. In fact, this thesis contains the following results: We generalize a unicity condition of three meromorphic functions given by Jank and Terglane in class A to the case of arbitrary q meromorphic functoins. An elementary proof of a conjecture of C. C. Yang is provided. We construct a class of meromorphic functions with exact two deficient values and their deficiencies are explicitly computed. We generalize the Nevanlinna`s five-value theorem to the cases that two meromorphic functions partially share either five or more values, or five or
more small functions. In each case, we formulate a way to measure how far these two meromorphic functions are from sharing either values or small functions, and use this measurement to prove a uniqueness theorem. Also, we prove some uniqueness theorems on entire functions that share a pair of values (a,-a) with their derivatives, which are reformulations of some important results about uniqueness of entire functions that share values with their derivatives. Finally, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ share four distinct values a_1, a_2, a_3, a_4 DM such that each a_i-point is either a (p,q)-fold or (q,p)-fold point of f and g, then (p,q) is either (1,2) or (1,3) and f, g are in some particular forms.		en_US
dc.description.tableofcontents 謝辭......................................................i

Abstract................................................iii

中文摘要..................................................iv

1 Introduction............................................1

2 Basic Theory of Value Distribution......................4

2.1 Poisson-Jensen`s Formula............................4

2.2 The Nevanlinna`s First Fundamental Theorem..........6

2.3 The Nevanlinna`s Second Fundamental Theorem.........8

2.4 The Estimation of S(r,f)............................9

2.5 Deficient Value of Meromorphic Functions...........12

2.6 Some Well-Known Results on Four Value Problem......13

3 Unicity of Meromorphic Functions of Class A............15

3.1 Introduction.......................................15

3.2 Some Facts About Meromorphic Functions of Class A..17

3.3 Main Results and Proofs............................18

3.4 A Conjecture.......................................20

4 On a Conjecture of C. C. Yang..........................22

4.1 Introduction.......................................22

4.2 Some Lemmas........................................24

4.3 Main Result and Proof..............................25

5 The Deficient Values of a Class of Meromorphic Functions
.......................................................28

5.1 Introduction.......................................28

5.2 The Deficient Values of Rational Functions.........30

5.3 The Proof of Theorem A.............................31

5.4 The Proof of Theorem B.............................35

6 Some Generalization of Nevanlinna`s Five-Values Theorem
.......................................................41

6.1 Introduction.......................................41

6.2 Meromorphic Functions Partially Share Values.......43

6.3 Meromorphic Functions Partially Share Small Functions
...................................................45

7 On the Uniqueness of Entire Functions and Their
Derivatives............................................49

7.1 Introduction.......................................49

7.2 Lemmas and Known Results...........................50

7.3 Main Results and Proofs............................52

8 Some Results on Meromorphic Functions Sharing Four Values
DM.....................................................65

8.1 Introduction.......................................65

8.2 Key Examples and Facts.............................66

8.3 Main Result and Proof..............................68

References...............................................69		zh_TW
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dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0093751501en_US
dc.subject (關鍵詞) 值分佈理論zh_TW
dc.subject (關鍵詞) 半純函數zh_TW
dc.subject (關鍵詞) value distribution theoryen_US
dc.subject (關鍵詞) meromorphic functionen_US
dc.title (題名) 半純函數的唯一性zh_TW
dc.title (題名) Some Results on the Uniqueness of Meromorphic Functionsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] W. W. Adams and E. G. Straus, Non-Archimedian analyticzh_TW
dc.relation.reference (參考文獻) functions taking the same values at the same points,zh_TW
dc.relation.reference (參考文獻) Ill. J. Math., 15 (1971), 418-424.zh_TW
dc.relation.reference (參考文獻) [2] G. Brosch, Eindeutigkeitssatze fur meromorphezh_TW
dc.relation.reference (參考文獻) funktionen, Thesis, Technical University of Aachen,zh_TW
dc.relation.reference (參考文獻) 1989.zh_TW
dc.relation.reference (參考文獻) [3] J. Clunie, On integral and meromorphic functions,zh_TW
dc.relation.reference (參考文獻) J. London Math. Soc., 36 (1962), 17-27.zh_TW
dc.relation.reference (參考文獻) [4] C. T. Chuang and C. C. Yang, Fixed points andzh_TW
dc.relation.reference (參考文獻) factorization theory of meromorphic functions, Pekingzh_TW
dc.relation.reference (參考文獻) Univ. Press, 1988.zh_TW
dc.relation.reference (參考文獻) [5] W. Doeringer, Exceptional value of differentialzh_TW
dc.relation.reference (參考文獻) polynomial, Pacific J. Math., 98 (1982), 55-62.zh_TW
dc.relation.reference (參考文獻) [6] G. Frank and W. Ohlenroth, Meromorphe funktionen, diezh_TW
dc.relation.reference (參考文獻) mit einer ihrer ableitungen werte teilen, Complexzh_TW
dc.relation.reference (參考文獻) Variables, 6 (1986), 23-37.zh_TW
dc.relation.reference (參考文獻) [7] F. Gross, Factorizatioin of meromorphic functions, U.zh_TW
dc.relation.reference (參考文獻) S. Government Printing Office, Washington, D. C.,1972.zh_TW
dc.relation.reference (參考文獻) [8] G. G. Gundersen, Meromorphic functions that share threezh_TW
dc.relation.reference (參考文獻) or four values, J. London Math. Soc., 20 (1979),zh_TW
dc.relation.reference (參考文獻) 457-466.zh_TW
dc.relation.reference (參考文獻) [9] G. G. Gundersen, Meromorphic functions that share finitezh_TW
dc.relation.reference (參考文獻) values with their derivative, J. Math. Anal. Appl.,zh_TW
dc.relation.reference (參考文獻) 75 (1980), 441-446.zh_TW
dc.relation.reference (參考文獻) [10] G. G. Gundersen, Meromorphic functions that share fourzh_TW
dc.relation.reference (參考文獻) values, Transactions of the American Mathematicalzh_TW
dc.relation.reference (參考文獻) Society, 277(2) (1983), 545-567.zh_TW
dc.relation.reference (參考文獻) [11] G. G. Gundersen and L. Z. Yang, Entire functions thatzh_TW
dc.relation.reference (參考文獻) share one value with one or two of their derivatives,zh_TW
dc.relation.reference (參考文獻) J. Math. Anal. Appl., 223 (1998), 88-95.zh_TW
dc.relation.reference (參考文獻) [12] W. K. Hayman, Meromorphic functions, Clarendon Press,zh_TW
dc.relation.reference (參考文獻) Oxford, 1964.zh_TW
dc.relation.reference (參考文獻) [13] D. Hans and S. Gerald, Zur charakterisierung vonzh_TW
dc.relation.reference (參考文獻) polynomen durch ihre Null-und Einsstellen, Arch.zh_TW
dc.relation.reference (參考文獻) Math., 48 (1987), 337-342.zh_TW
dc.relation.reference (參考文獻) [14] G. Jank and N. Terglane, Meromorphic functions sharingzh_TW
dc.relation.reference (參考文獻) three values, Math. Pannonica, 2 (1990), 37-46.zh_TW
dc.relation.reference (參考文獻) [15] P. Li, Entire functions that share one value with theirzh_TW
dc.relation.reference (參考文獻) linear differential polynomials, Kodai Math. J., 22zh_TW
dc.relation.reference (參考文獻) (1999), 446-457.zh_TW
dc.relation.reference (參考文獻) [16] P. Li and C. C. Yang, Uniqueness theorems on entirezh_TW
dc.relation.reference (參考文獻) functions and their derivatives, J. Math. Anal. Appl.,zh_TW
dc.relation.reference (參考文獻) 253 (2001), 50-57.zh_TW
dc.relation.reference (參考文獻) [17] Y. Li and J. Qiao, The uniqueness of meromorphiczh_TW
dc.relation.reference (參考文獻) functions concerning small functions, Sci. China Ser.zh_TW
dc.relation.reference (參考文獻) A, 43(6) (2000), 581-590.zh_TW
dc.relation.reference (參考文獻) [18] E. Mues, Meromorphic functions sharing four values,zh_TW
dc.relation.reference (參考文獻) Complex Variables, 12 (1989), 169-179.zh_TW
dc.relation.reference (參考文獻) [19] E. Mues, G. Jank and L. Volkmann, Meromorphezh_TW
dc.relation.reference (參考文獻) funktionen, die mit ihrer ersten und zweiten ableitungzh_TW
dc.relation.reference (參考文獻) einen endichen wert teilen, Complex Variables Theoryzh_TW
dc.relation.reference (參考文獻) Appl. 6(1986), 51-71.zh_TW
dc.relation.reference (參考文獻) [20] E. Mues and N. Steinmetz, Meromorphe funktionen, diezh_TW
dc.relation.reference (參考文獻) mit ihrer abelitung werte teilen, Manuscripta Math.zh_TW
dc.relation.reference (參考文獻) 29 (1979), 195-206.zh_TW
dc.relation.reference (參考文獻) [21] H. Milloux, Les fonctions meromorphes et leurszh_TW
dc.relation.reference (參考文獻) derivees, Paris, 1940.zh_TW
dc.relation.reference (參考文獻) [22] S. S. Miller, Complex analysis: Proceedings of the SUNYzh_TW
dc.relation.reference (參考文獻) Brockport Conference, Dekker, New York and Basel,zh_TW
dc.relation.reference (參考文獻) 1978, p.169.zh_TW
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dc.relation.reference (參考文獻) polynomials, Proc. Amer. Math. Soc., 82 (1981),zh_TW
dc.relation.reference (參考文獻) 183-187.zh_TW
dc.relation.reference (參考文獻) [24] K. Ninno and M. Ozawa, Deficiencies of an entirezh_TW
dc.relation.reference (參考文獻) algebroid function, Kodai Math. Sem. Rep., 22 (1970),zh_TW
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dc.relation.reference (參考文獻) [25] R. Nevanlinna, Le theoreme de Picard-Borel et lazh_TW
dc.relation.reference (參考文獻) theorie des fonctions meromorphes, Gauthiers-Villars,zh_TW
dc.relation.reference (參考文獻) Paris, 1929.zh_TW
dc.relation.reference (參考文獻) [26] R. Nevanlinna, Einige eindueutigkeitssatze in derzh_TW
dc.relation.reference (參考文獻) theorie der mermorphen funktionen, Acta Math., 48zh_TW
dc.relation.reference (參考文獻) (1926), 367-391.zh_TW
dc.relation.reference (參考文獻) [27] E. Picard. Memoire sur les fonctions entieres, Ann.zh_TW
dc.relation.reference (參考文獻) Ecole. Norm., 9(1880), 145-166.zh_TW
dc.relation.reference (參考文獻) [28] G. Polya. On an integral function of an integralzh_TW
dc.relation.reference (參考文獻) function, J. London Math. Soc., 1(1926), p.12.zh_TW
dc.relation.reference (參考文獻) [29] L. Ruble and C. C. Yang, Values shared by entirezh_TW
dc.relation.reference (參考文獻) functions and their derivatives, Complex Analysis,zh_TW
dc.relation.reference (參考文獻) Kentucky, 1976 (Berlin),Springer-Verlag, 1977, 101-103.zh_TW
dc.relation.reference (參考文獻) [30] M. Reinders, Eindeutigkeitssatze fur meromprphezh_TW
dc.relation.reference (參考文獻) Funktionen, die vier Werte teilen, PhD thesis,zh_TW
dc.relation.reference (參考文獻) Universitat Hannover, 1990.zh_TW
dc.relation.reference (參考文獻) [31] M. Reinders, Eindeutigkeitssatze fur meromorphezh_TW
dc.relation.reference (參考文獻) funktionen, die vier werte teilen, Mitt. Math. Sem.zh_TW
dc.relation.reference (參考文獻) Giessen, 200 (1991), 15-38.zh_TW
dc.relation.reference (參考文獻) [32] M. Reinders, A new example of meromorphic functionszh_TW
dc.relation.reference (參考文獻) sharing four values and a uniqueness theorem, Complexzh_TW
dc.relation.reference (參考文獻) Variables, 18 (1992), 213-221.zh_TW
dc.relation.reference (參考文獻) [33] N. Steinmetz, Eine Verallgemeinerung des zweitenzh_TW
dc.relation.reference (參考文獻) Nevanlinnaschen Hauptsatzes, J. Reine Angew. Math.,zh_TW
dc.relation.reference (參考文獻) 368 (1986) 134-141.zh_TW
dc.relation.reference (參考文獻) [34] S. P. Wang, On meromorphic functions that share fourzh_TW
dc.relation.reference (參考文獻) values, J. Math. Anal. Appl., 173 (1993), 359-369.zh_TW
dc.relation.reference (參考文獻) [35] H. X. Yi and C. C. Yang, Uniqueness theory ofzh_TW
dc.relation.reference (參考文獻) meromorphic functions, Pure and Applied Math.zh_TW
dc.relation.reference (參考文獻) Monographs No. 32, Science Press, Beijing, 1995.zh_TW
dc.relation.reference (參考文獻) [36] C. C. Yang, Some problems on polynomyals andzh_TW
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