【作者】 林珊华；

【导师】 林伟川； 潘飚； 吴桂荣；

【作者基本信息】 福建师范大学 ， 应用数学， 2006， 硕士

【摘要】 亚纯函数唯一性理论是值分布理论的一个重要研究方向。国内外许多数学家，如E．Mues，F．Gross，G．G．Gundersen，G．Frank，N．Steinmetz，W．Bergweiler，I．Laine，C．C．Yang，熊庆来，杨乐，仪洪勋和顾永兴等人，对亚纯函数的唯一性问题进行了大量深入的研究。本文主要是在“CM”与“IM”之间定义了权弱分担的概念，并采用权弱分担的思想讨论亚纯函数与其导数或齐次微分多项式分担小函数的唯一性问题，某些类型的微分多项式分担一个小函数的唯一性问题。全文共分如下四个部分： 第一部分为绪论。我们介绍Nevanlinna理论与亚纯函数唯一性理论的发展，以及亚纯函数唯一性理论中的几个结果。 在第二部分中，我们主要讨论亚纯函数与其k阶导数f^（k）权弱分担一个小函数的唯一性问题，回答了Kit-wing Yu提出的四个问题，并且推广了Kit-wing Yu，I．Lahiri和A．Sarkar，L．P．Liu和顾永兴等人的结果。 在第三部分中，我们主要讨论亚纯函数与其齐次微分多项式权弱分担两个小函数的唯一性问题，获得了两个唯一性定理，从而推广了L．A．Rubel，C．C．Yang，李效敏等人的结果。 在第四部分中，我们主要讨论某些类型的微分多项式权弱分担一个小函数的唯一性问题，推广了方明亮，仪洪勋，林伟川，熊维玲和Seiki Mori等人的结果。此外，给出的例子表明结果是精确的。更多还原

【Abstract】 The uniqueness theory of meromorphic functions is an important subject in value distribution theory. Many mathematicians such as E. Mues, F. Gross, G. G. Gundersen, G. Frank, N. Steinmetz, W. Bergweiler, I. Laine, C. C. Yang, Q. L. Xiong, L. Yang, H. X. Yi and Y. X. Gu had done a lot of work on the research of the uniqueness theory. In this paper, we mainly introduce the definition of weakly-weighted sharing which is between "CM" and "IM". Using the notion of weakly-weighted sharing, we study the uniqueness problems on a meromorphic function and its derivative or homogeneous differential polynomials sharing small functions, the uniqueness problems on some certain types of differential polynomials sharing one small function. This paper is divided into four sections:In section one, we mainly introduce the development of Nevanlinna theory and the uniqueness theory of meromorphic functions, and some results about the uniqueness theory of meromorphic functions.In section two, we mainly deal with the uniqueness problems on a meromorphic function and its kth order derivative f^（k） weakly-weighted sharing one small function. As consequences, we are able to answer four questions posed by Kit-wing Yu. Our results sharpen the results of Kit-wing Yu, I. Lahiri and A. Sarkar, L. P. Liu and Y. X. Gu.In section three, we are mainly concerned with the uniqueness problems of a meromorphic function and its homogeneous differential polynomials weakly-weighted sharing two small functions. We obtain two theorems, which improve some former results given by L. A. Rubel, C. C. Yang and X. M. Li.In section four, we are mainly devoted to the uniqueness problems on some certain types of differential polynomials weakly-weighted sharing one small function, which generalize the results of M. L. Fang, H. X. Yi, W. C. Lin, W. L. Xiong, Seiki Mori. Moreover, the example shows the results are sharp.更多还原