Uniqueness Of Meromorphic Functions Sharing Values With Their Derivatives And Some Related Problems

The value distribution theory founded by Rolf Nevanlinna in the 1920's. Usually, we called Nevanlinna theory in honor of him. Nevanlinna theory can be seen the most important achievements in the preceding century to understand the properties of meromorphic functions. This theory is composed of two main theorems, which are called Nevanlinna's first and second main theorems that had been significant breakthroughs in the development of the classical function theory, since the latter generalizes and extends the Picard's first theorem greatly, and hence it denoted the beginning of the theory of meromorphic functions. Moreover, Nevanlinna theory and its extensive has numerous applications in some fields of mathematics, for example, potential theory, complex difference equations, normal family, several complex variables and so on.The foundation of the normal family theory was laid by P. Montel in the early twentieth century. Let F be a family of meromorphic functions in a domain D (?) C. We say that F is normal in D if every sequence{fn}n (?) F contains a subsequence which converges spherically and uniformly on compact subsets of D, see [61]. Later on, with the in-depth study of Nevanlinna theory, normal family theory has developed rapidly. Many mathematicians have studied deeply and paid close attention to it. Prof. Q. L. Xiong, Prof. C. T. Chuang, Prof. L. Yang and Prof. G. H. Zhang, etc, made contributions to the introduction and development of the theory in China. The research on the normal family theory of meromorphic functions is a very active international subject in recent decades, especially with the foundation of Zalcman-Pang's Lemma. After that, a lot of elegant results were given by many mathematicians. In this period, many Chinese mathematicians, such as Prof. Y. X. Gu, Prof. H. H. Chen, Prof. X. C. Pang, Prof. M. L. Fang, etc, have made remarkable works and contributions to the development of the normal family theory in the world.The present thesis involves some results of the author that investigate the unicity of meromophic functions sharing value with their derivatives and study the existence of fixed points of the derivatives of solutions of complex linear differential equations in the unit disc, under the guidance of my supervisor Prof. H. X. Yi. It consists of four parts and the matters are explained as below.In Chapter 1, we introduce the general background of Nevanlinna Theory, the development of the Normal Family Theory, and the last section is a the very important theorem in this paper:Wiman-Valiron theorem.In Chapter 2, we investigate the uniqueness problem of entire functions that share polynomials or two transcendental meromorphic functions with their deriva-tives. By estimating the size ofρn's in the famous Zalcman-Pang's lemma and using the theory of normal family, we deduce this kind of functions have finite order, which is an important property. And then, we obtain some uniqueness the-orems, which improve the results given by Rubel and Yang [60], Li and Yi [47]. Meanwhile, some examples show that the conditions of the result are necessary. In fact, we obtained the following result.Theorem 0.1. Let R1 and R2 be two functions and R2((?)-R1,0), Let f(z) be a nonconstant meromorphic function with, finitely many poles. If f(z) = R1(?) f'(z)= R1, f(z) = R2(?) f'(z) = R2, f and R1 have no common poles and the order of R1 is less than the order of f, then one of the following cases must occur:(1) f≡f'.(2) f = R2+Ceλz and (λ-1)R1=λR2-R2' here C,λare two nonzero constants. In fact, R1, R2 are two polynomials.We also study the uniqueness problem of entire functions concerning their linear differential polynomial and obtain some results which improve some known theorems of J. L. Zhang [78] and Feng Lii [52]. We obtain:Theorem 0.2. Let f be a nonconstant meromorphic function with finitely many poles and all zeros of f have multiplicity at least k+1, and letα(z)= P(z)eQ(z), here P(z)(?) 0 and Q(z) are two polynomials. If then f(z) is of finite order.Theorem 0.3. Let f be a transcendental entire function whose zeros have mul-tiplicity at least k+1, and let Q(z)(?) 0 be a polynomial. If f and L(f) share Q(z) CM, then f≡L(f).In Chapter 3, we deal with the uniqueness problems of meromorphic func-tions that share a small function with its derivative and improve some results of Yang [71], Yu [74], Lahiri [44], and Zhang [77], also answer some questions of T. D. Zhang and W. R. Lu [80] as follows.Theorem 0.4. Let k(≥1), n(≥1), m(≥1) be integers and f be a non-constant meromorphic function. Also let a(z)((?) 0,∞) be a small function with respect to f. If fn and [f(k)]m share a(z) IM, and for 0<λ< 1, r∈I, and I is a set of infinite linear measure. Then for some constant c∈C\{0}.Theorem 0.5. Let k(≥1), n(≥1), m(≥1) be integers and f be a non-constant meromorphic function. Also let a(z)((?)0,∞) be a small function with respect to f. If fn and [f(k)]m share a(z) IM and In the finial Chapter 4. We do some research on the question of the exis-tence of fixed points of the derivatives of solutions of complex linear differential equations in the unit disc. This work improves some very recent results of T. B. Cao, see [7].