The Problem Of Value Sharing Of Meromorphic Functions And Their Derivatives

Value distribution theory of meromorphic functions, was due to R.Nevanlinna in 1920's, and in geometric form by L.Ahlfors about a decade later, is one of the most important achievements in the preceding century to understand the prop-erties of meromorphic functions. The theory is composed of two main theorems, which are called Nevanlinna's first and second theorems that had been significant breakthroughs in the development of the classic function theory, since the Nevan-linna's second theorem generalizes and extends Picard's first theorem greatly, and hence it denoted the beginning of the theory of meromorphic functions. Since then, Neanlinna theory has been well developed in itself and widely applied to the researches of the uniqueness of meromorphic functions,normal families, com-plex dynamics and differential equations etc. Meanwhile, in view of the beauty of Nevanliina theory, many outstanding mathematicians founded and developed the value distribution theory of meromorphic mappings over certain manifolds and p-adic field.The present thesis involves some results of the author that investigate the problem of value sharing of meromorphic functions and their derivatives and some problem in p-adic field, under the guidance of supervisor professor Peichu Hu.The dissertation is structured as follows.In Chapter 1, we describe the basic Nevanlinna theory in complex number field and p-adic field.In Chapter 2, we consider the problem of value sharing of meromorphic functions and their first derivatives, mainly improve the theorem 2 in J.T. Li and H.X. Yi[16], we get theorem 2.1 by replacing the values a, b in theorem 2 by two meromorphic functions. Next, we apply a theorem given by J. Wang[26] recently, replace "f(z)=R1(?)f'(z)=R1" in theorem 2.1 by " f(z)=R1→f'(z)=R1" and then get theorem 2.2. Furthermore, as a kind of generalization and complement, we get theorem 2.3 and 2.4.In Chapter 3. we consider the problem of value sharing of meromorphic functions and their k-th derivatives, mainly improve the theorem 1 in C. Wu and J.T. Li[28], we get theorem 3.1 by replacing the values a, b,d in theorem 1 by three polynomials. In Chapter 4, we consider a more wide certain type of function equation: and in p-adic field we give a necessary condition of the existence of meromorphic solutions i.e. theorem 4.1, which generalizes some results of N. Toda[23]. K.W.Yu and C.C.Yang[33].