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題名 半純函數與其導數之值分佈
On The Value Distribution Of Meromorphic Functions With Their Derivatives
作者 歐姿君
Ou, Tze Chun
貢獻者 陳天進
Chen, Ten Ging
歐姿君
Ou, Tze Chun
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 13:48:01 (UTC+8)
摘要 Haymen猜測:對任意的超越半純函數 f(z)，f`(z)f(z)^n 取所有值無窮多次，其中至多只有一個例外值。這個著名的猜測，大部分的情形已被證明是正確的。另外，Hayman 證明 f`(z)-af(z)^n 取所有有限值無窮多次
，其中 a 為一複數且 n≧5 的正整數。在本篇論文裡，我們將探討以小函數為係數的半純函數微分多項式之值分佈問題。並將Hayman的結果推廣至 f^{k}(z)f(z)^n 與 f^{k}(z)-af(z)^n 的情形。同時，我們也證明一些
A類半純函數與其導數的值分佈結果。
A famous conjecture of Hayman says that if f(z) is a transcendental meromorphic function, then f`(z)f(z)^n assumes all finite values except possibly zero infinitely often. The conjecture was solved in most cases. Another result of Hayman says that f`(z)-af(z)^n, where n≧5 and a is a complex number, assumes all finite values infinitely often. In this thesis, we will study the value distribution of some differential polynomial in a meromorphic function with small functions as coefficents. In fact, we will generalize Hayman`s results to the cases f^(k)(z)f(z)^n and f^(k)(z)-af(z)^n. Also, the value distribution of meromorphic functions of class A with their derivatives are obtained.
參考文獻 [1] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic
function of finite order, Rev. Mat. Iber., 11 (1995), 355-373.
[2] H. H. Chen and M. L. Fang, On the value distribution of fnf0, Sci. China Ser.
A, 38 (1995), 789-798.
[3] J. Clunie, On integral and meromorphic function, J. London Math. Soc., 37
(1962), 17-27.
[4] J. Clunie, On a result of Hayman, J. London Math. Soc., 42 (1967), 389-392.
[5] P. Csillag, ¨ Uber ganze funktionen, welche drei nicht verschwindende ableitungen
besitzen, Math. Ann., 110 (1935), 745-752.
[6] G. Frank, Eine vermutung von Hayman ¨uber nullstellen meromorpher funktion,
Math. Z., 149 (1976), 29-36.
[7] G. Frank, W. Hennekemper and G. Polloczek, ¨ Uber die nullstellen meromorpher
funktionen und deren ableitungen, Math. Ann., no.2 225 (1977), 145-154.
[8] W. K. Hayman, Picard values of meromorphic functions and their derivatives,
Ann. Math., 70 (1959), 9-42.
[9] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964.
[10] W. K. Hayman, Reseach Problems in Function Theory, London: Athlone Press,
1967.
[11] J. K. Langley, Proof of a conjecture of Hayman concerning f and f00, J. London
Math. Soc., no.2 48 (1993), 500-514.
[12] E. Mues, ¨ Uber ein problem von Hayman, Math. Z., 164 (1979), 239-259.
[13] E. Mues, Meromorphic functions sharing four values, Complex Variables, 12
(1989), 169–179.
[14] W. Saxer, Sur les valeurs exceptionelles des d´eriv´ees successives des fonctions
meromorphes, C. R. Acad. Sci. Paris, 182 (1926), 831-833.
[15] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions,
Kluwer Academic Publishers, 2003.
[16] C. C. Yang and C. T. Chuang, Fixed points and factorization theory of meromorphic
functions, Peking Univ. Press, 1988.
[17] L. Zalcman, On some problems of Hayman, preprint (Bar-Ilan University).
[18] L. Yang, Value distribution theory, Berlin Heidelberg: Springer-Verlag, Beijing:
Science Press, 1993.
[19] F. Gross, Factorizatioin of meromorphic functions, U. S. Government Printing
Office, Washington, D. C.,1972.
[20] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Pure
and Applied Math. Monographs No. 32, Science Press, Beijing, 1995.
[21] R. Nevanlinna, Le th´eor"eme de Picard-Borel et la th´eorie des fonctions
m´eromorphes, Gauthiers-Villars, Paris, 1929.
[22] H. Milloux, Les fonctions m´eromorphes et leurs d´eriv´ees, Paris, 1940.
[23] K. Y. Chen, Some Results on the Uniqueness of Meromorphic Functions, PHD
thesis, National Chengchi University, 2007.
描述 碩士
國立政治大學
應用數學研究所
94751008
96
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0094751008
資料類型 thesis
dc.contributor.advisor 陳天進zh_TW
dc.contributor.advisor Chen, Ten Gingen_US
dc.contributor.author (Authors) 歐姿君zh_TW
dc.contributor.author (Authors) Ou, Tze Chunen_US
dc.creator (作者) 歐姿君zh_TW
dc.creator (作者) Ou, Tze Chunen_US
dc.date (日期) 2007en_US
dc.date.accessioned 17-Sep-2009 13:48:01 (UTC+8)-
dc.date.available 17-Sep-2009 13:48:01 (UTC+8)-
dc.date.issued (上傳時間) 17-Sep-2009 13:48:01 (UTC+8)-
dc.identifier (Other Identifiers) G0094751008en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/32586-
dc.description (描述) 碩士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) 94751008zh_TW
dc.description (描述) 96zh_TW
dc.description.abstract (摘要) Haymen猜測:對任意的超越半純函數 f(z)，f`(z)f(z)^n 取所有值無窮多次，其中至多只有一個例外值。這個著名的猜測，大部分的情形已被證明是正確的。另外，Hayman 證明 f`(z)-af(z)^n 取所有有限值無窮多次
，其中 a 為一複數且 n≧5 的正整數。在本篇論文裡，我們將探討以小函數為係數的半純函數微分多項式之值分佈問題。並將Hayman的結果推廣至 f^{k}(z)f(z)^n 與 f^{k}(z)-af(z)^n 的情形。同時，我們也證明一些
A類半純函數與其導數的值分佈結果。		zh_TW
dc.description.abstract (摘要) A famous conjecture of Hayman says that if f(z) is a transcendental meromorphic function, then f`(z)f(z)^n assumes all finite values except possibly zero infinitely often. The conjecture was solved in most cases. Another result of Hayman says that f`(z)-af(z)^n, where n≧5 and a is a complex number, assumes all finite values infinitely often. In this thesis, we will study the value distribution of some differential polynomial in a meromorphic function with small functions as coefficents. In fact, we will generalize Hayman`s results to the cases f^(k)(z)f(z)^n and f^(k)(z)-af(z)^n. Also, the value distribution of meromorphic functions of class A with their derivatives are obtained.en_US
dc.description.tableofcontents 謝辭......................................................i

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

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

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

2 Basic Theory of Nevanlinna`s Value Distribution Theory......................4

3.Some Lemmas.............................14

4.Value Distribution of Meromorphic Functions in class A with Their Derivatives..............16

5.Value Distribution of Meromorphic Functions with Their Derivatives...............................21

6.References.............................................................33		zh_TW
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dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0094751008en_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) 半純函數與其導數之值分佈zh_TW
dc.title (題名) On The Value Distribution Of Meromorphic Functions With Their Derivativesen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphiczh_TW
dc.relation.reference (參考文獻) function of finite order, Rev. Mat. Iber., 11 (1995), 355-373.zh_TW
dc.relation.reference (參考文獻) [2] H. H. Chen and M. L. Fang, On the value distribution of fnf0, Sci. China Ser.zh_TW
dc.relation.reference (參考文獻) A, 38 (1995), 789-798.zh_TW
dc.relation.reference (參考文獻) [3] J. Clunie, On integral and meromorphic function, J. London Math. Soc., 37zh_TW
dc.relation.reference (參考文獻) (1962), 17-27.zh_TW
dc.relation.reference (參考文獻) [4] J. Clunie, On a result of Hayman, J. London Math. Soc., 42 (1967), 389-392.zh_TW
dc.relation.reference (參考文獻) [5] P. Csillag, ¨ Uber ganze funktionen, welche drei nicht verschwindende ableitungenzh_TW
dc.relation.reference (參考文獻) besitzen, Math. Ann., 110 (1935), 745-752.zh_TW
dc.relation.reference (參考文獻) [6] G. Frank, Eine vermutung von Hayman ¨uber nullstellen meromorpher funktion,zh_TW
dc.relation.reference (參考文獻) Math. Z., 149 (1976), 29-36.zh_TW
dc.relation.reference (參考文獻) [7] G. Frank, W. Hennekemper and G. Polloczek, ¨ Uber die nullstellen meromorpherzh_TW
dc.relation.reference (參考文獻) funktionen und deren ableitungen, Math. Ann., no.2 225 (1977), 145-154.zh_TW
dc.relation.reference (參考文獻) [8] W. K. Hayman, Picard values of meromorphic functions and their derivatives,zh_TW
dc.relation.reference (參考文獻) Ann. Math., 70 (1959), 9-42.zh_TW
dc.relation.reference (參考文獻) [9] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964.zh_TW
dc.relation.reference (參考文獻) [10] W. K. Hayman, Reseach Problems in Function Theory, London: Athlone Press,zh_TW
dc.relation.reference (參考文獻) 1967.zh_TW
dc.relation.reference (參考文獻) [11] J. K. Langley, Proof of a conjecture of Hayman concerning f and f00, J. Londonzh_TW
dc.relation.reference (參考文獻) Math. Soc., no.2 48 (1993), 500-514.zh_TW
dc.relation.reference (參考文獻) [12] E. Mues, ¨ Uber ein problem von Hayman, Math. Z., 164 (1979), 239-259.zh_TW
dc.relation.reference (參考文獻) [13] E. Mues, Meromorphic functions sharing four values, Complex Variables, 12zh_TW
dc.relation.reference (參考文獻) (1989), 169–179.zh_TW
dc.relation.reference (參考文獻) [14] W. Saxer, Sur les valeurs exceptionelles des d´eriv´ees successives des fonctionszh_TW
dc.relation.reference (參考文獻) meromorphes, C. R. Acad. Sci. Paris, 182 (1926), 831-833.zh_TW
dc.relation.reference (參考文獻) [15] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions,zh_TW
dc.relation.reference (參考文獻) Kluwer Academic Publishers, 2003.zh_TW
dc.relation.reference (參考文獻) [16] C. C. Yang and C. T. Chuang, Fixed points and factorization theory of meromorphiczh_TW
dc.relation.reference (參考文獻) functions, Peking Univ. Press, 1988.zh_TW
dc.relation.reference (參考文獻) [17] L. Zalcman, On some problems of Hayman, preprint (Bar-Ilan University).zh_TW
dc.relation.reference (參考文獻) [18] L. Yang, Value distribution theory, Berlin Heidelberg: Springer-Verlag, Beijing:zh_TW
dc.relation.reference (參考文獻) Science Press, 1993.zh_TW
dc.relation.reference (參考文獻) [19] F. Gross, Factorizatioin of meromorphic functions, U. S. Government Printingzh_TW
dc.relation.reference (參考文獻) Office, Washington, D. C.,1972.zh_TW
dc.relation.reference (參考文獻) [20] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Purezh_TW
dc.relation.reference (參考文獻) and Applied Math. Monographs No. 32, Science Press, Beijing, 1995.zh_TW
dc.relation.reference (參考文獻) [21] R. Nevanlinna, Le th´eor"eme de Picard-Borel et la th´eorie des fonctionszh_TW
dc.relation.reference (參考文獻) m´eromorphes, Gauthiers-Villars, Paris, 1929.zh_TW
dc.relation.reference (參考文獻) [22] H. Milloux, Les fonctions m´eromorphes et leurs d´eriv´ees, Paris, 1940.zh_TW
dc.relation.reference (參考文獻) [23] K. Y. Chen, Some Results on the Uniqueness of Meromorphic Functions, PHDzh_TW
dc.relation.reference (參考文獻) thesis, National Chengchi University, 2007.zh_TW