Please use this identifier to cite or link to this item: https://ah.nccu.edu.tw/handle/140.119/32586


Title: 半純函數與其導數之值分佈
On The Value Distribution Of Meromorphic Functions With Their Derivatives
Authors: 歐姿君
Ou, Tze Chun
Contributors: 陳天進
Chen, Ten Ging
歐姿君
Ou, Tze Chun
Keywords: 值分佈理論
半純函數
value distribution theory
meromorphic function
Date: 2007
Issue Date: 2009-09-17 13:48:01 (UTC+8)
Abstract: 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.
Reference: [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.
Description: 碩士
國立政治大學
應用數學研究所
94751008
96
Source URI: http://thesis.lib.nccu.edu.tw/record/#G0094751008
Data Type: thesis
Appears in Collections:[應用數學系] 學位論文

Files in This Item:

File Description SizeFormat
100801.pdf46KbAdobe PDF593View/Open
100802.pdf25KbAdobe PDF644View/Open
100803.pdf103KbAdobe PDF657View/Open
100804.pdf26KbAdobe PDF652View/Open
100805.pdf67KbAdobe PDF680View/Open
100806.pdf47KbAdobe PDF738View/Open
100807.pdf88KbAdobe PDF1205View/Open
100808.pdf49KbAdobe PDF843View/Open
100809.pdf62KbAdobe PDF840View/Open
100810.pdf107KbAdobe PDF691View/Open
100811.pdf42KbAdobe PDF785View/Open


All items in 學術集成 are protected by copyright, with all rights reserved.


社群 sharing