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| Title: | 半純函數的唯一性
Some Results on the Uniqueness of Meromorphic Functions |
| Authors: | 陳耿彥
Chen, Keng-Yan |
| Contributors: | 陳天進
Chen, Ten-Ging
陳耿彥
Chen, Keng-Yan |
| Keywords: | 值分佈理論
半純函數
value distribution theory
meromorphic function |
| Date: | 2007 |
| Issue Date: | 2009-09-17 13:47:35 (UTC+8) |
| Abstract: | 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. |
| Reference: | [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. |
| Description: | 博士
國立政治大學
應用數學研究所
93751501
96 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0093751501 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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